2016

• Paper 1, Section I, D

  Analysis I

  Part IA, 2016

  What does it mean to say that a sequence of real numbers $\left(x_{n}\right)$ converges to $x$ ? Suppose that $\left(x_{n}\right)$ converges to $x$. Show that the sequence $\left(y_{n}\right)$ given by

  $$y_{n}=\frac{1}{n} \sum_{i=1}^{n} x_{i}$$

  also converges to $x$.

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• Paper 1, Section I, F

  Analysis I

  Part IA, 2016

  Let $a_{n}$ be the number of pairs of integers $(x, y) \in \mathbb{Z}^{2}$ such that $x^{2}+y^{2} \leqslant n^{2}$. What is the radius of convergence of the series $\sum_{n=0}^{\infty} a_{n} z^{n}$ ? [You may use the comparison test, provided you state it clearly.]

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• Paper 1, Section II, E

  Analysis I

  Part IA, 2016

  State the Bolzano-Weierstrass theorem. Use it to show that a continuous function $f:[a, b] \rightarrow \mathbb{R}$ attains a global maximum; that is, there is a real number $c \in[a, b]$ such that $f(c) \geqslant f(x)$ for all $x \in[a, b]$.

  A function $f$ is said to attain a local maximum at $c \in \mathbb{R}$ if there is some $\varepsilon>0$ such that $f(c) \geqslant f(x)$ whenever $|x-c|<\varepsilon$. Suppose that $f: \mathbb{R} \rightarrow \mathbb{R}$ is twice differentiable, and that $f^{\prime \prime}(x)<0$ for all $x \in \mathbb{R}$. Show that there is at most one $c \in \mathbb{R}$ at which $f$ attains a local maximum.

  If there is a constant $K<0$ such that $f^{\prime \prime}(x)<K$ for all $x \in \mathbb{R}$, show that $f$ attains a global maximum. [Hint: if $g^{\prime}(x)<0$ for all $x \in \mathbb{R}$, then $g$ is decreasing.]

  Must $f: \mathbb{R} \rightarrow \mathbb{R}$ attain a global maximum if we merely require $f^{\prime \prime}(x)<0$ for all $x \in \mathbb{R} ?$ Justify your answer.

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• Paper 1, Section II, E

  Analysis I

  Part IA, 2016

  Let $f: \mathbb{R} \rightarrow \mathbb{R}$. We say that $x \in \mathbb{R}$ is a real root of $f$ if $f(x)=0$. Show that if $f$ is differentiable and has $k$ distinct real roots, then $f^{\prime}$ has at least $k-1$ real roots. [Rolle's theorem may be used, provided you state it clearly.]

  Let $p(x)=\sum_{i=1}^{n} a_{i} x^{d_{i}}$ be a polynomial in $x$, where all $a_{i} \neq 0$ and $d_{i+1}>d_{i}$. (In other words, the $a_{i}$ are the nonzero coefficients of the polynomial, arranged in order of increasing power of $x$.) The number of sign changes in the coefficients of $p$ is the number of $i$ for which $a_{i} a_{i+1}<0$. For example, the polynomial $x^{5}-x^{3}-x^{2}+1$ has 2 sign changes. Show by induction on $n$ that the number of positive real roots of $p$ is less than or equal to the number of sign changes in its coefficients.

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• Paper 1, Section II, D

  Analysis I

  Part IA, 2016

  If $\left(x_{n}\right)$ and $\left(y_{n}\right)$ are sequences converging to $x$ and $y$ respectively, show that the sequence $\left(x_{n}+y_{n}\right)$ converges to $x+y$.

  If $x_{n} \neq 0$ for all $n$ and $x \neq 0$, show that the sequence $\left(\frac{1}{x_{n}}\right)$ converges to $\frac{1}{x}$.

  (a) Find $\lim _{n \rightarrow \infty}\left(\sqrt{n^{2}+n}-n\right)$.

  (b) Determine whether $\sum_{n=1}^{\infty} \frac{\sqrt{n+1}-\sqrt{n}}{\sqrt{n}}$ converges.

  Justify your answers.

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• Paper 1, Section II, 12F

  Analysis I

  Part IA, 2016

  Let $f:[0,1] \rightarrow \mathbb{R}$ satisfy $|f(x)-f(y)| \leqslant|x-y|$ for all $x, y \in[0,1]$.

  Show that $f$ is continuous and that for all $\varepsilon>0$, there exists a piecewise constant function $g$ such that

  $$\sup _{x \in[0,1]}|f(x)-g(x)| \leqslant \varepsilon .$$

  For all integers $n \geqslant 1$, let $u_{n}=\int_{0}^{1} f(t) \cos (n t) d t$. Show that the sequence $\left(u_{n}\right)$ converges to 0 .

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• Paper 2, Section I, A

  Differential Equations

  Part IA, 2016

  (a) Find the solution of the differential equation

  $$y^{\prime \prime}-y^{\prime}-6 y=0$$

  that is bounded as $x \rightarrow \infty$ and satisfies $y=1$ when $x=0$.

  (b) Solve the difference equation

  $$\left(y_{n+1}-2 y_{n}+y_{n-1}\right)-\frac{h}{2}\left(y_{n+1}-y_{n-1}\right)-6 h^{2} y_{n}=0 .$$

  Show that if $0<h \ll 1$, the solution that is bounded as $n \rightarrow \infty$ and satisfies $y_{0}=1$ is approximately $(1-2 h)^{n}$.

  (c) By setting $x=n h$, explain the relation between parts (a) and (b).

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• Paper 2, Section II, A

  Differential Equations

  Part IA, 2016

  The function $y(x)$ satisfies

  $$y^{\prime \prime}+p(x) y^{\prime}+q(x) y=0$$

  What does it mean to say that the point $x=0$ is (i) an ordinary point and (ii) a regular singular point of this differential equation? Explain what is meant by the indicial equation at a regular singular point. What can be said about the nature of the solutions in the neighbourhood of a regular singular point in the different cases that arise according to the values of the roots of the indicial equation?

  State the nature of the point $x=0$ of the equation

  $$x y^{\prime \prime}+(x-m+1) y^{\prime}-(m-1) y=0$$

  Set $y(x)=x^{\sigma} \sum_{n=0}^{\infty} a_{n} x^{n}$, where $a_{0} \neq 0$, and find the roots of the indicial equation.

  (a) Show that one solution of $(*)$ with $m \neq 0,-1,-2, \cdots$ is

  $$y(x)=x^{m}\left(1+\sum_{n=1}^{\infty} \frac{(-1)^{n} x^{n}}{(m+n)(m+n-1) \cdots(m+1)}\right)$$

  and find a linearly independent solution in the case when $m$ is not an integer.

  (b) If $m$ is a positive integer, show that $(*)$ has a polynomial solution.

  (c) What is the form of the general solution of $(*)$ in the case $m=0$ ? [You do not need to find the general solution explicitly.]

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• Paper 2, Section $I$, A

  Differential Equations

  Part IA, 2016

  (a) For each non-negative integer $n$ and positive constant $\lambda$, let

  $$I_{n}(\lambda)=\int_{0}^{\infty} x^{n} e^{-\lambda x} d x$$

  By differentiating $I_{n}$ with respect to $\lambda$, find its value in terms of $n$ and $\lambda$.

  (b) By making the change of variables $x=u+v, y=u-v$, transform the differential equation

  $$\frac{\partial^{2} f}{\partial x \partial y}=1$$

  into a differential equation for $g$, where $g(u, v)=f(x, y)$.

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• Paper 2, Section II, A

  Differential Equations

  Part IA, 2016

  (a) Find and sketch the solution of

  $$y^{\prime \prime}+y=\delta(x-\pi / 2)$$

  where $\delta$ is the Dirac delta function, subject to $y(0)=1$ and $y^{\prime}(0)=0$.

  (b) A bowl of soup, which Sam has just warmed up, cools down at a rate equal to the product of a constant $k$ and the difference between its temperature $T(t)$ and the temperature $T_{0}$ of its surroundings. Initially the soup is at temperature $T(0)=\alpha T_{0}$, where $\alpha>2$.

  (i) Write down and solve the differential equation satisfied by $T(t)$.

  (ii) At time $t_{1}$, when the temperature reaches half of its initial value, Sam quickly adds some hot water to the soup, so the temperature increases instantaneously by $\beta$, where $\beta>\alpha T_{0} / 2$. Find $t_{1}$ and $T(t)$ for $t>t_{1}$.

  (iii) Sketch $T(t)$ for $t>0$.

  (iv) Sam wants the soup to be at temperature $\alpha T_{0}$ at time $t_{2}$, where $t_{2}>t_{1}$. What value of $\beta$ should Sam choose to achieve this? Give your answer in terms of $\alpha$, $k, t_{2}$ and $T_{0}$.

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• Paper 2, Section II, $6 A$

  Differential Equations

  Part IA, 2016

  (a) The function $y(x)$ satisfies

  $$y^{\prime \prime}+p(x) y^{\prime}+q(x) y=0$$

  (i) Define the Wronskian $W(x)$ of two linearly independent solutions $y_{1}(x)$ and $y_{2}(x)$. Derive a linear first-order differential equation satisfied by $W(x)$.

  (ii) Suppose that $y_{1}(x)$ is known. Use the Wronskian to write down a first-order differential equation for $y_{2}(x)$. Hence express $y_{2}(x)$ in terms of $y_{1}(x)$ and $W(x)$.

  (b) Verify that $y_{1}(x)=\cos \left(x^{\gamma}\right)$ is a solution of

  $$a x^{\alpha} y^{\prime \prime}+b x^{\alpha-1} y^{\prime}+y=0,$$

  where $a, b, \alpha$ and $\gamma$ are constants, provided that these constants satisfy certain conditions which you should determine.

  Use the method that you described in part (a) to find a solution which is linearly independent of $y_{1}(x)$.

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• Paper 2, Section II, A

  Differential Equations

  Part IA, 2016

  (a) By considering eigenvectors, find the general solution of the equations

  $$\begin{aligned} &\frac{d x}{d t}=2 x+5 y \\ &\frac{d y}{d t}=-x-2 y \end{aligned}$$

  and show that it can be written in the form

  $$\left(\begin{array}{l} x \\ y \end{array}\right)=\alpha\left(\begin{array}{c} 5 \cos t \\ -2 \cos t-\sin t \end{array}\right)+\beta\left(\begin{array}{c} 5 \sin t \\ \cos t-2 \sin t \end{array}\right)$$

  where $\alpha$ and $\beta$ are constants.

  (b) For any square matrix $M$, $\exp (M)$ is defined by

  $$\exp (M)=\sum_{n=0}^{\infty} \frac{M^{n}}{n !}$$

  Show that if $M$ has constant elements, the vector equation $\frac{d \mathbf{x}}{d t}=M \mathbf{x}$ has a solution $\mathbf{x}=\exp (M t) \mathbf{x}_{0}$, where $\mathbf{x}_{0}$ is a constant vector. Hence solve $(\dagger)$ and show that your solution is consistent with the result of part (a).

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• Paper 4, Section I, B

  Dynamics and Relativity

  Part IA, 2016

  With the help of definitions or equations of your choice, determine the dimensions, in terms of mass $(M)$, length $(L)$, time $(T)$ and charge $(Q)$, of the following quantities:

  (i) force;

  (ii) moment of a force (i.e. torque);

  (iii) energy;

  (iv) Newton's gravitational constant $G$;

  (v) electric field $\mathbf{E}$;

  (vi) magnetic field $\mathbf{B}$;

  (vii) the vacuum permittivity $\epsilon_{0}$.

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• Paper 4, Section $I$, B

  Dynamics and Relativity

  Part IA, 2016

  The radial equation of motion of a particle moving under the influence of a central force is

  $$\ddot{r}-\frac{h^{2}}{r^{3}}=-k r^{n}$$

  where $h$ is the angular momentum per unit mass of the particle, $n$ is a constant, and $k$ is a positive constant.

  Show that circular orbits with $r=a$ are possible for any positive value of $a$, and that they are stable to small perturbations that leave $h$ unchanged if $n>-3$.

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• Paper 4, Section II, B

  Dynamics and Relativity

  Part IA, 2016

  (a) A rocket, moving non-relativistically, has speed $v(t)$ and mass $m(t)$ at a time $t$ after it was fired. It ejects mass with constant speed $u$ relative to the rocket. Let the total momentum, at time $t$, of the system (rocket and ejected mass) in the direction of the motion of the rocket be $P(t)$. Explain carefully why $P(t)$ can be written in the form

  $$P(t)=m(t) v(t)-\int_{0}^{t}(v(\tau)-u) \frac{d m(\tau)}{d \tau} d \tau$$

  If the rocket experiences no external force, show that

  $$m \frac{d v}{d t}+u \frac{d m}{d t}=0$$

  Derive the expression corresponding to $(*)$ for the total kinetic energy of the system at time $t$. Show that kinetic energy is not necessarily conserved.

  (b) Explain carefully how $(*)$ should be modified for a rocket moving relativistically, given that there are no external forces. Deduce that

  $$\frac{d(m \gamma v)}{d t}=\left(\frac{v-u}{1-u v / c^{2}}\right) \frac{d(m \gamma)}{d t}$$

  where $\gamma=\left(1-v^{2} / c^{2}\right)^{-\frac{1}{2}}$ and hence that

  $$m \gamma^{2} \frac{d v}{d t}+u \frac{d m}{d t}=0$$

  (c) Show that $(\dagger)$ and $(\dagger)$ agree in the limit $c \rightarrow \infty$. Briefly explain the fact that kinetic energy is not conserved for the non-relativistic rocket, but relativistic energy is conserved for the relativistic rocket.

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• Paper 4, Section II, B

  Dynamics and Relativity

  Part IA, 2016

  A particle of unit mass moves with angular momentum $h$ in an attractive central force field of magnitude $\frac{k}{r^{2}}$, where $r$ is the distance from the particle to the centre and $k$ is a constant. You may assume that the equation of its orbit can be written in plane polar coordinates in the form

  $$r=\frac{\ell}{1+e \cos \theta}$$

  where $\ell=\frac{h^{2}}{k}$ and $e$ is the eccentricity. Show that the energy of the particle is

  $$\frac{h^{2}\left(e^{2}-1\right)}{2 \ell^{2}}$$

  A comet moves in a parabolic orbit about the Sun. When it is at its perihelion, a distance $d$ from the Sun, and moving with speed $V$, it receives an impulse which imparts an additional velocity of magnitude $\alpha V$ directly away from the Sun. Show that the eccentricity of its new orbit is $\sqrt{1+4 \alpha^{2}}$, and sketch the two orbits on the same axes.

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• Paper 4, Section II, B

  Dynamics and Relativity

  Part IA, 2016

  (a) Alice travels at constant speed $v$ to Alpha Centauri, which is at distance $d$ from Earth. She then turns around (taking very little time to do so), and returns at speed $v$. Bob stays at home. By how much has Bob aged during the journey? By how much has Alice aged? [No justification is required.]

  Briefly explain what is meant by the twin paradox in this context. Why is it not a paradox?

  (b) Suppose instead that Alice's world line is given by

  $$-c^{2} t^{2}+x^{2}=c^{2} t_{0}^{2},$$

  where $t_{0}$ is a positive constant. Bob stays at home, at $x=\alpha c t_{0}$, where $\alpha>1$. Alice and Bob compare their ages on both occasions when they meet. By how much does Bob age? Show that Alice ages by $2 t_{0} \cosh ^{-1} \alpha$.

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• Paper 4, Section II, B

  Dynamics and Relativity

  Part IA, 2016

  State what the vectors $\mathbf{a}, \mathbf{r}, \mathbf{v}$ and $\boldsymbol{\omega}$ represent in the following equation:

  $$\mathbf{a}=\mathbf{g}-2 \boldsymbol{\omega} \times \mathbf{v}-\boldsymbol{\omega} \times(\boldsymbol{\omega} \times \mathbf{r})$$

  where $\mathbf{g}$ is the acceleration due to gravity.

  Assume that the radius of the Earth is $6 \times 10^{6} \mathrm{~m}$, that $|\mathrm{g}|=10 \mathrm{~ms}^{-2}$, and that there are $9 \times 10^{4}$ seconds in a day. Use these data to determine roughly the order of magnitude of each term on the right hand side of $(*)$ in the case of a particle dropped from a point at height $20 \mathrm{~m}$ above the surface of the Earth.

  Taking again $|\mathbf{g}|=10 \mathrm{~ms}^{-2}$, find the time $T$ of the particle's fall in the absence of rotation.

  Use a suitable approximation scheme to show that

  $$\mathbf{R} \approx \mathbf{R}_{0}-\frac{1}{3} \boldsymbol{\omega} \times \mathbf{g} T^{3}-\frac{1}{2} \boldsymbol{\omega} \times\left(\boldsymbol{\omega} \times \mathbf{R}_{0}\right) T^{2},$$

  where $\mathbf{R}$ is the position vector of the point at which the particle lands, and $\mathbf{R}_{0}$ is the position vector of the point at which the particle would have landed in the absence of rotation.

  The particle is dropped at latitude $45^{\circ}$. Find expressions for the approximate northerly and easterly displacements of $\mathbf{R}$ from $\mathbf{R}_{0}$ in terms of $\omega, g, R_{0}$ (the magnitudes of $\boldsymbol{\omega}, \mathbf{g}$ and $\mathbf{R}_{0}$, respectively), and $T$. You should ignore the curvature of the Earth's surface.

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• Paper 3, Section I, D

  Groups

  Part IA, 2016

  Let $G$ be a group, and let $H$ be a subgroup of $G$. Show that the following are equivalent.

  (i) $a^{-1} b^{-1} a b \in H$ for all $a, b \in G$.

  (ii) $H$ is a normal subgroup of $G$ and $G / H$ is abelian.

  Hence find all abelian quotient groups of the dihedral group $D_{10}$ of order 10 .

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• Paper 3, Section I, D

  Groups

  Part IA, 2016

  State and prove Lagrange's theorem.

  Let $p$ be an odd prime number, and let $G$ be a finite group of order $2 p$ which has a normal subgroup of order 2 . Show that $G$ is a cyclic group.

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• Paper 3, Section II, D

  Groups

  Part IA, 2016

  For each of the following, either give an example or show that none exists.

  (i) A non-abelian group in which every non-trivial element has order $2 .$

  (ii) A non-abelian group in which every non-trivial element has order 3 .

  (iii) An element of $S_{9}$ of order 18 .

  (iv) An element of $S_{9}$ of order 20 .

  (v) A finite group which is not isomorphic to a subgroup of an alternating group.

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• Paper 3, Section II, D

  Groups

  Part IA, 2016

  Define the sign, $\operatorname{sgn}(\sigma)$, of a permutation $\sigma \in S_{n}$ and prove that it is well defined. Show that the function $\operatorname{sgn}: S_{n} \rightarrow\{1,-1\}$ is a homomorphism.

  Show that there is an injective homomorphism $\psi: G L_{2}(\mathbb{Z} / 2 \mathbb{Z}) \rightarrow S_{4}$ such that $\operatorname{sgn} \circ \psi$ is non-trivial.

  Show that there is an injective homomorphism $\phi: S_{n} \rightarrow G L_{n}(\mathbb{R})$ such that $\operatorname{det}(\phi(\sigma))=\operatorname{sgn}(\sigma) .$

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• Paper 3, Section II, $7 \mathrm{D}$

  Groups

  Part IA, 2016

  State and prove the orbit-stabiliser theorem.

  Let $p$ be a prime number, and $G$ be a finite group of order $p^{n}$ with $n \geqslant 1$. If $N$ is a non-trivial normal subgroup of $G$, show that $N \cap Z(G)$ contains a non-trivial element.

  If $H$ is a proper subgroup of $G$, show that there is a $g \in G \backslash H$ such that $g^{-1} H g=H$.

  [You may use Lagrange's theorem, provided you state it clearly.]

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• Paper 3, Section II, D

  Groups

  Part IA, 2016

  Define the Möbius group $\mathcal{M}$ and its action on the Riemann sphere $\mathbb{C}_{\infty}$. [You are not required to verify the group axioms.] Show that there is a surjective group homomorphism $\phi: S L_{2}(\mathbb{C}) \rightarrow \mathcal{M}$, and find the kernel of $\phi .$

  Show that if a non-trivial element of $\mathcal{M}$ has finite order, then it fixes precisely two points in $\mathbb{C}_{\infty}$. Hence show that any finite abelian subgroup of $\mathcal{M}$ is either cyclic or isomorphic to $C_{2} \times C_{2}$.

  [You may use standard properties of the Möbius group, provided that you state them clearly.]

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• Paper 4, Section I, E

  Numbers and Sets

  Part IA, 2016

  Find a pair of integers $x$ and $y$ satisfying $17 x+29 y=1$. What is the smallest positive integer congruent to $17^{138}$ modulo 29 ?

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• Paper 4, Section I, E

  Numbers and Sets

  Part IA, 2016

  Explain the meaning of the phrase least upper bound; state the least upper bound property of the real numbers. Use the least upper bound property to show that a bounded, increasing sequence of real numbers converges.

  Suppose that $a_{n}, b_{n} \in \mathbb{R}$ and that $a_{n} \geqslant b_{n}>0$ for all $n$. If $\sum_{n=1}^{\infty} a_{n}$ converges, show that $\sum_{n=1}^{\infty} b_{n}$ converges.

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• Paper 4, Section II, E

  Numbers and Sets

  Part IA, 2016

  (a) Let $S$ be a set. Show that there is no bijective map from $S$ to the power set of $S$. Let $\mathcal{T}=\left\{\left(x_{n}\right) \mid x_{i} \in\{0,1\}\right.$ for all $\left.i \in \mathbb{N}\right\}$ be the set of sequences with entries in $\{0,1\} .$ Show that $\mathcal{T}$ is uncountable.

  (b) Let $A$ be a finite set with more than one element, and let $B$ be a countably infinite set. Determine whether each of the following sets is countable. Justify your answers.

  (i) $S_{1}=\{f: A \rightarrow B \mid f$ is injective $\}$.

  (ii) $S_{2}=\{g: B \rightarrow A \mid g$ is surjective $\}$.

  (iii) $S_{3}=\{h: B \rightarrow B \mid h$ is bijective $\}$.

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• Paper 4, Section II, $6 \mathrm{E}$

  Numbers and Sets

  Part IA, 2016

  Suppose that $a, b \in \mathbb{Z}$ and that $b=b_{1} b_{2}$, where $b_{1}$ and $b_{2}$ are relatively prime and greater than 1. Show that there exist unique integers $a_{1}, a_{2}, n \in \mathbb{Z}$ such that $0 \leqslant a_{i}<b_{i}$ and

  $$\frac{a}{b}=\frac{a_{1}}{b_{1}}+\frac{a_{2}}{b_{2}}+n$$

  Now let $b=p_{1}^{n_{1}} \cdots p_{k}^{n_{k}}$ be the prime factorization of $b$. Deduce that $\frac{a}{b}$ can be written uniquely in the form

  $$\frac{a}{b}=\frac{q_{1}}{p_{1}^{n_{1}}}+\cdots+\frac{q_{k}}{p_{k}^{n_{k}}}+n$$

  where $0 \leqslant q_{i}<p_{i}^{n_{i}}$ and $n \in \mathbb{Z}$. Express $\frac{a}{b}=\frac{1}{315}$ in this form.

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• Paper 4, Section II, $7 \mathrm{E}$

  Numbers and Sets

  Part IA, 2016

  State the inclusion-exclusion principle.

  Let $A=\left(a_{1}, a_{2}, \ldots, a_{n}\right)$ be a string of $n$ digits, where $a_{i} \in\{0,1, \ldots, 9\}$. We say that the string $A$ has a run of length $k$ if there is some $j \leqslant n-k+1$ such that either $a_{j+i} \equiv a_{j}+i(\bmod 10)$ for all $0 \leqslant i<k$ or $a_{j+i} \equiv a_{j}-i(\bmod 10)$ for all $0 \leqslant i<k$. For example, the strings

  $$(\underline{0,1,2}, 8,4,9),(3, \underline{9,8,7}, 4,8) \text { and }(3, \underline{1,0,9}, 4,5)$$

  all have runs of length 3 (underlined), but no run in $(3,1,2,1,1,2)$ has length $>2$. How many strings of length 6 have a run of length $\geqslant 3$ ?

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• Paper 4, Section II, 8E

  Numbers and Sets

  Part IA, 2016

  Define the binomial coefficient $\left(\begin{array}{c}n \\ m\end{array}\right)$. Prove directly from your definition that

  $$(1+z)^{n}=\sum_{m=0}^{n}\left(\begin{array}{c} n \\ m \end{array}\right) z^{m}$$

  for any complex number $z$.

  (a) Using this formula, or otherwise, show that

  $$\sum_{k=0}^{3 n}(-3)^{k}\left(\begin{array}{l} 6 n \\ 2 k \end{array}\right)=2^{6 n}$$

  (b) By differentiating, or otherwise, evaluate $\sum_{m=0}^{n} m\left(\begin{array}{c}n \\ m\end{array}\right)$.

  Let $S_{r}(n)=\sum_{m=0}^{n}(-1)^{m} m^{r}\left(\begin{array}{c}n \\ m\end{array}\right)$, where $r$ is a non-negative integer. Show that $S_{r}(n)=0$ for $r<n$. Evaluate $S_{n}(n)$.

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• Paper 2, Section I, F

  Probability

  Part IA, 2016

  Let $X_{1}, \ldots, X_{n}$ be independent random variables, all with uniform distribution on $[0,1]$. What is the probability of the event $\left\{X_{1}>X_{2}>\cdots>X_{n-1}>X_{n}\right\}$ ?

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• Paper 2, Section I, $4 \mathrm{~F}$

  Probability

  Part IA, 2016

  Define the moment-generating function $m_{Z}$ of a random variable $Z$. Let $X_{1}, \ldots, X_{n}$ be independent and identically distributed random variables with distribution $\mathcal{N}(0,1)$, and let $Z=X_{1}^{2}+\cdots+X_{n}^{2}$. For $\theta<1 / 2$, show that

  $$m_{Z}(\theta)=(1-2 \theta)^{-n / 2} .$$

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• Paper 2, Section II, F

  Probability

  Part IA, 2016

  For any positive integer $n$ and positive real number $\theta$, the Gamma distribution $\Gamma(n, \theta)$ has density $f_{\Gamma}$ defined on $(0, \infty)$ by

  $$f_{\Gamma}(x)=\frac{\theta^{n}}{(n-1) !} x^{n-1} e^{-\theta x} .$$

  For any positive integers $a$ and $b$, the Beta distribution $B(a, b)$ has density $f_{B}$ defined on $(0,1)$ by

  $$f_{B}(x)=\frac{(a+b-1) !}{(a-1) !(b-1) !} x^{a-1}(1-x)^{b-1}$$

  Let $X$ and $Y$ be independent random variables with respective distributions $\Gamma(n, \theta)$ and $\Gamma(m, \theta)$. Show that the random variables $X /(X+Y)$ and $X+Y$ are independent and give their distributions.

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• Paper 2, Section II, F

  Probability

  Part IA, 2016

  We randomly place $n$ balls in $m$ bins independently and uniformly. For each $i$ with $1 \leqslant i \leqslant m$, let $B_{i}$ be the number of balls in bin $i$.

  (a) What is the distribution of $B_{i}$ ? For $i \neq j$, are $B_{i}$ and $B_{j}$ independent?

  (b) Let $E$ be the number of empty bins, $C$ the number of bins with two or more balls, and $S$ the number of bins with exactly one ball. What are the expectations of $E, C$ and $S$ ?

  (c) Let $m=a n$, for an integer $a \geqslant 2$. What is $\mathbb{P}(E=0)$ ? What is the limit of $\mathbb{E}[E] / m$ when $n \rightarrow \infty$ ?

  (d) Instead, let $n=d m$, for an integer $d \geqslant 2$. What is $\mathbb{P}(C=0)$ ? What is the limit of $\mathbb{E}[C] / m$ when $n \rightarrow \infty$ ?

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• Paper 2, Section II, F

  Probability

  Part IA, 2016

  Let $X$ be a non-negative random variable such that $\mathbb{E}\left[X^{2}\right]>0$ is finite, and let $\theta \in[0,1]$.

  (a) Show that

  $$\mathbb{E}[X \mathbb{I}[\{X>\theta \mathbb{E}[X]\}]] \geqslant(1-\theta) \mathbb{E}[X]$$

  (b) Let $Y_{1}$ and $Y_{2}$ be random variables such that $\mathbb{E}\left[Y_{1}^{2}\right]$ and $\mathbb{E}\left[Y_{2}^{2}\right]$ are finite. State and prove the Cauchy-Schwarz inequality for these two variables.

  (c) Show that

  $$\mathbb{P}(X>\theta \mathbb{E}[X]) \geqslant(1-\theta)^{2} \frac{\mathbb{E}[X]^{2}}{\mathbb{E}\left[X^{2}\right]}$$

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• Paper 2, Section II, F

  Probability

  Part IA, 2016

  A random graph with $n$ nodes $v_{1}, \ldots, v_{n}$ is drawn by placing an edge with probability $p$ between $v_{i}$ and $v_{j}$ for all distinct $i$ and $j$, independently. A triangle is a set of three distinct nodes $v_{i}, v_{j}, v_{k}$ that are all connected: there are edges between $v_{i}$ and $v_{j}$, between $v_{j}$ and $v_{k}$ and between $v_{i}$ and $v_{k}$.

  (a) Let $T$ be the number of triangles in this random graph. Compute the maximum value and the expectation of $T$.

  (b) State the Markov inequality. Show that if $p=1 / n^{\alpha}$, for some $\alpha>1$, then $\mathbb{P}(T=0) \rightarrow 1$ when $n \rightarrow \infty$

  (c) State the Chebyshev inequality. Show that if $p$ is such that $\operatorname{Var}[T] / \mathbb{E}[T]^{2} \rightarrow 0$ when $n \rightarrow \infty$, then $\mathbb{P}(T=0) \rightarrow 0$ when $n \rightarrow \infty$

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• Paper 3, Section I, C

  Vector Calculus

  Part IA, 2016

  State the chain rule for the derivative of a composition $t \mapsto f(\mathbf{X}(t))$, where $f: \mathbb{R}^{n} \rightarrow \mathbb{R}$ and $\mathbf{X}: \mathbb{R} \rightarrow \mathbb{R}^{n}$ are smooth $.$

  Consider parametrized curves given by

  $$\mathbf{x}(t)=(x(t), y(t))=(a \cos t, a \sin t) .$$

  Calculate the tangent vector $\frac{d \mathbf{x}}{d t}$ in terms of $x(t)$ and $y(t)$. Given that $u(x, y)$ is a smooth function in the upper half-plane $\left\{(x, y) \in \mathbb{R}^{2} \mid y>0\right\}$ satisfying

  $$x \frac{\partial u}{\partial y}-y \frac{\partial u}{\partial x}=u$$

  deduce that

  $$\frac{d}{d t} u(x(t), y(t))=u(x(t), y(t))$$

  If $u(1,1)=10$, find $u(-1,1)$.

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• Paper 3, Section I, C

  Vector Calculus

  Part IA, 2016

  If $\mathbf{v}=\left(v_{1}, v_{2}, v_{3}\right)$ and $\mathbf{w}=\left(w_{1}, w_{2}, w_{3}\right)$ are vectors in $\mathbb{R}^{3}$, show that $T_{i j}=v_{i} w_{j}$ defines a rank 2 tensor. For which choices of the vectors $\mathbf{v}$ and $\mathbf{w}$ is $T_{i j}$ isotropic?

  Write down the most general isotropic tensor of rank 2 .

  Prove that $\epsilon_{i j k}$ defines an isotropic rank 3 tensor.

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• Paper 3, Section II, C

  Vector Calculus

  Part IA, 2016

  What is a conservative vector field on $\mathbb{R}^{n}$ ?

  State Green's theorem in the plane $\mathbb{R}^{2}$.

  (a) Consider a smooth vector field $\mathbf{V}=(P(x, y), Q(x, y))$ defined on all of $\mathbb{R}^{2}$ which satisfies

  $$\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}=0$$

  By considering

  $$F(x, y)=\int_{0}^{x} P\left(x^{\prime}, 0\right) d x^{\prime}+\int_{0}^{y} Q\left(x, y^{\prime}\right) d y^{\prime}$$

  or otherwise, show that $\mathbf{V}$ is conservative.

  (b) Now let $\mathbf{V}=(1+\cos (2 \pi x+2 \pi y), 2+\cos (2 \pi x+2 \pi y))$. Show that there exists a smooth function $F(x, y)$ such that $\mathbf{V}=\nabla F$.

  Calculate $\int_{C} \mathbf{V} \cdot d \mathbf{x}$, where $C$ is a smooth curve running from $(0,0)$ to $(m, n) \in \mathbb{Z}^{2}$. Deduce that there does not exist a smooth function $F(x, y)$ which satisfies $\mathbf{V}=\nabla F$ and which is, in addition, periodic with period 1 in each coordinate direction, i.e. $F(x, y)=F(x+1, y)=F(x, y+1)$.

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• Paper 3, Section II, C

  Vector Calculus

  Part IA, 2016

  Define the Jacobian $J[\mathbf{u}]$ of a smooth mapping $\mathbf{u}: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}$. Show that if $\mathbf{V}$ is the vector field with components

  $$V_{i}=\frac{1}{3 !} \epsilon_{i j k} \epsilon_{a b c} \frac{\partial u_{a}}{\partial x_{j}} \frac{\partial u_{b}}{\partial x_{k}} u_{c}$$

  then $J[\mathbf{u}]=\nabla \cdot \mathbf{V}$. If $\mathbf{v}$ is another such mapping, state the chain rule formula for the derivative of the composition $\mathbf{w}(\mathbf{x})=\mathbf{u}(\mathbf{v}(\mathbf{x}))$, and hence give $J[\mathbf{w}]$ in terms of $J[\mathbf{u}]$ and $J[\mathbf{v}]$.

  Let $\mathbf{F}: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}$ be a smooth vector field. Let there be given, for each $t \in \mathbb{R}$, a smooth mapping $\mathbf{u}_{t}: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}$ such that $\mathbf{u}_{t}(\mathbf{x})=\mathbf{x}+t \mathbf{F}(\mathbf{x})+o(t)$ as $t \rightarrow 0$. Show that

  $$J\left[\mathbf{u}_{t}\right]=1+t Q(x)+o(t)$$

  for some $Q(x)$, and express $Q$ in terms of $\mathbf{F}$. Assuming now that $\mathbf{u}_{t+s}(\mathbf{x})=\mathbf{u}_{t}\left(\mathbf{u}_{s}(\mathbf{x})\right)$, deduce that if $\nabla \cdot \mathbf{F}=0$ then $J\left[\mathbf{u}_{t}\right]=1$ for all $t \in \mathbb{R}$. What geometric property of the mapping $\mathbf{x} \mapsto \mathbf{u}_{t}(\mathbf{x})$ does this correspond to?

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• Paper 3, Section II, C

  Vector Calculus

  Part IA, 2016

  (a) For smooth scalar fields $u$ and $v$, derive the identity

  $$\nabla \cdot(u \nabla v-v \nabla u)=u \nabla^{2} v-v \nabla^{2} u$$

  and deduce that

  $$\begin{aligned} \int_{\rho \leqslant|\mathbf{x}| \leqslant r}\left(v \nabla^{2} u-u \nabla^{2} v\right) d V=\int_{|\mathbf{x}|=r}\left(v \frac{\partial u}{\partial n}-u \frac{\partial v}{\partial n}\right) d S \\ &-\int_{|\mathbf{x}|=\rho}\left(v \frac{\partial u}{\partial n}-u \frac{\partial v}{\partial n}\right) d S \end{aligned}$$

  Here $\nabla^{2}$ is the Laplacian, $\frac{\partial}{\partial n}=\mathbf{n} \cdot \nabla$ where $\mathbf{n}$ is the unit outward normal, and $d S$ is the scalar area element.

  (b) Give the expression for $(\nabla \times \mathbf{V})_{i}$ in terms of $\epsilon_{i j k}$. Hence show that

  $$\nabla \times(\nabla \times \mathbf{V})=\nabla(\nabla \cdot \mathbf{V})-\nabla^{2} \mathbf{V}$$

  (c) Assume that if $\nabla^{2} \varphi=-\rho$, where $\varphi(\mathbf{x})=O\left(|\mathbf{x}|^{-1}\right)$ and $\nabla \varphi(\mathbf{x})=O\left(|\mathbf{x}|^{-2}\right)$ as $|\mathbf{x}| \rightarrow \infty$, then

  $$\varphi(\mathbf{x})=\int_{\mathbb{R}^{3}} \frac{\rho(\mathbf{y})}{4 \pi|\mathbf{x}-\mathbf{y}|} d V .$$

  The vector fields $\mathbf{B}$ and $\mathbf{J}$ satisfy

  $$\nabla \times \mathbf{B}=\mathbf{J}$$

  Show that $\nabla \cdot \mathbf{J}=0$. In the case that $\mathbf{B}=\nabla \times \mathbf{A}$, with $\nabla \cdot \mathbf{A}=0$, show that

  $$\mathbf{A}(\mathbf{x})=\int_{\mathbb{R}^{3}} \frac{\mathbf{J}(\mathbf{y})}{4 \pi|\mathbf{x}-\mathbf{y}|} d V$$

  and hence that

  $$\mathbf{B}(\mathbf{x})=\int_{\mathbb{R}^{3}} \frac{\mathbf{J}(\mathbf{y}) \times(\mathbf{x}-\mathbf{y})}{4 \pi|\mathbf{x}-\mathbf{y}|^{3}} d V$$

  Verify that $\mathbf{A}$ given by $(*)$ does indeed satisfy $\nabla \cdot \mathbf{A}=0$. [It may be useful to make a change of variables in the right hand side of $(*)$.]

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• Paper 3, Section II, C

  Vector Calculus

  Part IA, 2016

  (a) Let

  $$\mathbf{F}=(z, x, y)$$

  and let $C$ be a circle of radius $R$ lying in a plane with unit normal vector $(a, b, c)$. Calculate $\nabla \times \mathbf{F}$ and use this to compute $\oint_{C} \mathbf{F} \cdot d \mathbf{x}$. Explain any orientation conventions which you use.

  (b) Let $\mathbf{F}: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}$ be a smooth vector field such that the matrix with entries $\frac{\partial F_{j}}{\partial x_{i}}$ is symmetric. Prove that $\oint_{C} \mathbf{F} \cdot d \mathbf{x}=0$ for every circle $C \subset \mathbb{R}^{3}$.

  (c) Let $\mathbf{F}=\frac{1}{r}(x, y, z)$, where $r=\sqrt{x^{2}+y^{2}+z^{2}}$ and let $C$ be the circle which is the intersection of the sphere $(x-5)^{2}+(y-3)^{2}+(z-2)^{2}=1$ with the plane $3 x-5 y-z=2$. Calculate $\oint_{C} \mathbf{F} \cdot d \mathbf{x}$.

  (d) Let $\mathbf{F}$ be the vector field defined, for $x^{2}+y^{2}>0$, by

  $$\mathbf{F}=\left(\frac{-y}{x^{2}+y^{2}}, \frac{x}{x^{2}+y^{2}}, z\right)$$

  Show that $\nabla \times \mathbf{F}=\mathbf{0}$. Let $C$ be the curve which is the intersection of the cylinder $x^{2}+y^{2}=1$ with the plane $z=x+200$. Calculate $\oint_{C} \mathbf{F} \cdot d \mathbf{x}$.

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• Paper 1, Section I, A

  Vectors and Matrices

  Part IA, 2016

  Let $z \in \mathbb{C}$ be a solution of

  $$z^{2}+b z+1=0$$

  where $b \in \mathbb{R}$ and $|b| \leqslant 2$. For which values of $b$ do the following hold?

  (i) $\left|e^{z}\right|<1$.

  (ii) $\left|e^{i z}\right|=1$.

  (iii) $\operatorname{Im}(\cosh z)=0$.

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• Paper 1, Section I, C

  Vectors and Matrices

  Part IA, 2016

  Write down the general form of a $2 \times 2$ rotation matrix. Let $R$ be a real $2 \times 2$ matrix with positive determinant such that $|R \mathbf{x}|=|\mathbf{x}|$ for all $\mathbf{x} \in \mathbb{R}^{2}$. Show that $R$ is a rotation matrix.

  Let

  $$J=\left(\begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array}\right)$$

  Show that any real $2 \times 2$ matrix $A$ which satisfies $A J=J A$ can be written as $A=\lambda R$, where $\lambda$ is a real number and $R$ is a rotation matrix.

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• Paper 1, Section II, A

  Vectors and Matrices

  Part IA, 2016

  (a) Use suffix notation to prove that

  $$\mathbf{a} \cdot(\mathbf{b} \times \mathbf{c})=\mathbf{c} \cdot(\mathbf{a} \times \mathbf{b})$$

  (b) Show that the equation of the plane through three non-colinear points with position vectors $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$ is

  $$\mathbf{r} \cdot(\mathbf{a} \times \mathbf{b}+\mathbf{b} \times \mathbf{c}+\mathbf{c} \times \mathbf{a})=\mathbf{a} \cdot(\mathbf{b} \times \mathbf{c})$$

  where $\mathbf{r}$ is the position vector of a point in this plane.

  Find a unit vector normal to the plane in the case $\mathbf{a}=(2,0,1), \mathbf{b}=(0,4,0)$ and $\mathbf{c}=(1,-1,2)$.

  (c) Let $\mathbf{r}$ be the position vector of a point in a given plane. The plane is a distance $d$ from the origin and has unit normal vector $\mathbf{n}$, where $\mathbf{n} \cdot \mathbf{r} \geqslant 0$. Write down the equation of this plane.

  This plane intersects the sphere with centre at $\mathbf{p}$ and radius $q$ in a circle with centre at $\mathbf{m}$ and radius $\rho$. Show that

  $$\mathbf{m}-\mathbf{p}=\gamma \mathbf{n}$$

  Find $\gamma$ in terms of $q$ and $\rho$. Hence find $\rho$ in terms of $\mathbf{n}, d, \mathbf{p}$ and $q$.

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• Paper 1, Section II, $\mathbf{6 B}$

  Vectors and Matrices

  Part IA, 2016

  The $n \times n$ real symmetric matrix $M$ has eigenvectors of unit length $\mathbf{e}_{1}, \mathbf{e}_{2}, \ldots, \mathbf{e}_{n}$, with corresponding eigenvalues $\lambda_{1}, \lambda_{2}, \ldots, \lambda_{n}$, where $\lambda_{1}>\lambda_{2}>\cdots>\lambda_{n}$. Prove that the eigenvalues are real and that $\mathbf{e}_{a} \cdot \mathbf{e}_{b}=\delta_{a b}$.

  Let $\mathbf{x}$ be any (real) unit vector. Show that

  $$\mathbf{x}^{\mathrm{T}} M \mathrm{x} \leqslant \lambda_{1}$$

  What can be said about $\mathbf{x}$ if $\mathbf{x}^{\mathrm{T}} M \mathbf{x}=\lambda_{1} ?$

  Let $S$ be the set of all (real) unit vectors of the form

  $$\mathbf{x}=\left(0, x_{2}, \ldots, x_{n}\right)$$

  Show that $\alpha_{1} \mathbf{e}_{1}+\alpha_{2} \mathbf{e}_{2} \in S$ for some $\alpha_{1}, \alpha_{2} \in \mathbb{R}$. Deduce that

  $$\underset{\mathbf{x} \in S}{\operatorname{Max}} \mathbf{x}^{\mathrm{T}} M \mathbf{x} \geqslant \lambda_{2}$$

  The $(n-1) \times(n-1)$ matrix $A$ is obtained by removing the first row and the first column of $M$. Let $\mu$ be the greatest eigenvalue of $A$. Show that

  $$\lambda_{1} \geqslant \mu \geqslant \lambda_{2}$$

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• Paper 1, Section II, B

  Vectors and Matrices

  Part IA, 2016

  What does it mean to say that a matrix can be diagonalised? Given that the $n \times n$ real matrix $M$ has $n$ eigenvectors satisfying $\mathbf{e}_{a} \cdot \mathbf{e}_{b}=\delta_{a b}$, explain how to obtain the diagonal form $\Lambda$ of $M$. Prove that $\Lambda$ is indeed diagonal. Obtain, with proof, an expression for the trace of $M$ in terms of its eigenvalues.

  The elements of $M$ are given by

  $$M_{i j}= \begin{cases}0 & \text { for } i=j \\ 1 & \text { for } i \neq j\end{cases}$$

  Determine the elements of $M^{2}$ and hence show that, if $\lambda$ is an eigenvalue of $M$, then

  $$\lambda^{2}=(n-1)+(n-2) \lambda$$

  Assuming that $M$ can be diagonalised, give its diagonal form.

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• Paper 1, Section II, $8 \mathbf{C}$

  Vectors and Matrices

  Part IA, 2016

  (a) Show that the equations

  $$\begin{array}{r} 1+s+t=a \\ 1-s+t=b \\ 1-2 t=c \end{array}$$

  determine $s$ and $t$ uniquely if and only if $a+b+c=3$.

  Write the following system of equations

  $$\begin{aligned} &5 x+2 y-z=1+s+t \\ &2 x+5 y-z=1-s+t \\ &-x-y+8 z=1-2 t \end{aligned}$$

  in matrix form $A \mathbf{x}=\mathbf{b}$. Use Gaussian elimination to solve the system for $x, y$, and $z$. State a relationship between the rank and the kernel of a matrix. What is the rank and what is the kernel of $A$ ?

  For which values of $x, y$, and $z$ is it possible to solve the above system for $s$ and $t$ ?

  (b) Define a unitary $n \times n$ matrix. Let $A$ be a real symmetric $n \times n$ matrix, and let $I$ be the $n \times n$ identity matrix. Show that $|(A+i I) \mathbf{x}|^{2}=|A \mathbf{x}|^{2}+|\mathbf{x}|^{2}$ for arbitrary $\mathbf{x} \in \mathbb{C}^{n}$, where $|\mathbf{x}|^{2}=\sum_{j=1}^{n}\left|x_{j}\right|^{2}$. Find a similar expression for $|(A-i I) \mathbf{x}|^{2}$. Prove that $(A-i I)(A+i I)^{-1}$ is well-defined and is a unitary matrix.

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• Paper 3, Section I, G

  Analysis II

  Part IB, 2016

  (a) Let $X$ be a subset of $\mathbb{R}$. What does it mean to say that a sequence of functions $f_{n}: X \rightarrow \mathbb{R}(n \in \mathbb{N})$ is uniformly convergent?

  (b) Which of the following sequences of functions are uniformly convergent? Justify your answers.

  (i) $f_{n}:(0,1) \rightarrow \mathbb{R}, \quad f_{n}(x)=\frac{1-x^{n}}{1-x}$

  (ii) $f_{n}:(0,1) \rightarrow \mathbb{R}, \quad f_{n}(x)=\sum_{k=1}^{n} \frac{1}{k^{2}} x^{k}$.

  (iii) $f_{n}: \mathbb{R} \rightarrow \mathbb{R}$, $\quad f_{n}(x)=x / n$.

  (iv) $f_{n}:[0, \infty) \rightarrow \mathbb{R}, \quad f_{n}(x)=x e^{-n x}$.

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• Paper 4, Section I, G

  Analysis II

  Part IB, 2016

  (a) What does it mean to say that a mapping $f: X \rightarrow X$ from a metric space to itself is a contraction?

  (b) State carefully the contraction mapping theorem.

  (c) Let $\left(a_{1}, a_{2}, a_{3}\right) \in \mathbb{R}^{3}$. By considering the metric space $\left(\mathbb{R}^{3}, d\right)$ with

  $$d(x, y)=\sum_{i=1}^{3}\left|x_{i}-y_{i}\right|$$

  or otherwise, show that there exists a unique solution $\left(x_{1}, x_{2}, x_{3}\right) \in \mathbb{R}^{3}$ of the system of equations

  $$\begin{aligned} &x_{1}=a_{1}+\frac{1}{6}\left(\sin x_{2}+\sin x_{3}\right), \\ &x_{2}=a_{2}+\frac{1}{6}\left(\sin x_{1}+\sin x_{3}\right), \\ &x_{3}=a_{3}+\frac{1}{6}\left(\sin x_{1}+\sin x_{2}\right) . \end{aligned}$$

  comment

• Paper 2, Section I, G

  Analysis II

  Part IB, 2016

  (a) What does it mean to say that the function $f: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$ is differentiable at the point $x=\left(x_{1}, x_{2}, \ldots, x_{n}\right) \in \mathbb{R}^{n}$ ? Show from your definition that if $f$ is differentiable at $x$, then $f$ is continuous at $x$.

  (b) Suppose that there are functions $g_{j}: \mathbb{R} \rightarrow \mathbb{R}^{m}(1 \leqslant j \leqslant n)$ such that for every $x=\left(x_{1}, \ldots, x_{n}\right) \in \mathbb{R}^{n}$,

  $$f(x)=\sum_{j=1}^{n} g_{j}\left(x_{j}\right) .$$

  Show that $f$ is differentiable at $x$ if and only if each $g_{j}$ is differentiable at $x_{j}$.

  (c) Let $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ be given by

  $$f(x, y)=|x|^{3 / 2}+|y|^{1 / 2}$$

  Determine at which points $(x, y) \in \mathbb{R}^{2}$ the function $f$ is differentiable.

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• Paper 1, Section II, G

  Analysis II

  Part IB, 2016

  Let $(X, d)$ be a metric space.

  (a) What does it mean to say that $\left(x_{n}\right)_{n}$ is a Cauchy sequence in $X$ ? Show that if $\left(x_{n}\right)_{n}$ is a Cauchy sequence, then it converges if it contains a convergent subsequence.

  (b) Let $\left(x_{n}\right)_{n}$ be a Cauchy sequence in $X$.

  (i) Show that for every $m \geqslant 1$, the sequence $\left(d\left(x_{m}, x_{n}\right)\right)_{n}$ converges to some $d_{m} \in \mathbb{R}$.

  (ii) Show that $d_{m} \rightarrow 0$ as $m \rightarrow \infty$.

  (iii) Let $\left(y_{n}\right)_{n}$ be a subsequence of $\left(x_{n}\right)_{n}$. If $\ell, m$ are such that $y_{\ell}=x_{m}$, show that $d\left(y_{\ell}, y_{n}\right) \rightarrow d_{m}$ as $n \rightarrow \infty$.

  (iv) Show also that for every $m$ and $n$,

  $$d_{m}-d_{n} \leqslant d\left(x_{m}, x_{n}\right) \leqslant d_{m}+d_{n}$$

  (v) Deduce that $\left(x_{n}\right)_{n}$ has a subsequence $\left(y_{n}\right)_{n}$ such that for every $m$ and $n$,

  $$d\left(y_{m+1}, y_{m}\right) \leqslant \frac{1}{3} d\left(y_{m}, y_{m-1}\right)$$

  and

  $$d\left(y_{m+1}, y_{n+1}\right) \leqslant \frac{1}{2} d\left(y_{m}, y_{n}\right)$$

  (c) Suppose that every closed subset $Y$ of $X$ has the property that every contraction mapping $Y \rightarrow Y$ has a fixed point. Prove that $X$ is complete.

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• Paper 4, Section II, G

  Analysis II

  Part IB, 2016

  (a) Let $V$ be a real vector space. What does it mean to say that two norms on $V$ are Lipschitz equivalent? Prove that every norm on $\mathbb{R}^{n}$ is Lipschitz equivalent to the Euclidean norm. Hence or otherwise, show that any linear map from $\mathbb{R}^{n}$ to $\mathbb{R}^{m}$ is continuous.

  (b) Let $f: U \rightarrow V$ be a linear map between normed real vector spaces. We say that $f$ is bounded if there exists a constant $C$ such that for all $u \in U,\|f(u)\| \leqslant C\|u\|$. Show that $f$ is bounded if and only if $f$ is continuous.

  (c) Let $\ell^{2}$ denote the space of sequences $\left(x_{n}\right)_{n \geqslant 1}$ of real numbers such that $\sum_{n \geqslant 1} x_{n}^{2}$ is convergent, with the norm $\left\|\left(x_{n}\right)_{n}\right\|=\left(\sum_{n \geqslant 1} x_{n}^{2}\right)^{1 / 2}$. Let $e_{m} \in \ell^{2}$ be the sequence $e_{m}=\left(x_{n}\right)_{n}$ with $x_{m}=1$ and $x_{n}=0$ if $n \neq m$. Let $w$ be the sequence $\left(2^{-n}\right)_{n}$. Show that the subset $\{w\} \cup\left\{e_{m} \mid m \geqslant 1\right\}$ is linearly independent. Let $V \subset \ell^{2}$ be the subspace it spans, and consider the linear map $f: V \rightarrow \mathbb{R}$ defined by

  $$f(w)=1, \quad f\left(e_{m}\right)=0 \quad \text { for all } m \geqslant 1 .$$

  Is $f$ continuous? Justify your answer.

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• Paper 3, Section II, G

  Analysis II

  Part IB, 2016

  Let $X$ be a metric space.

  (a) What does it mean to say that a function $f: X \rightarrow \mathbb{R}$ is uniformly continuous? What does it mean to say that $f$ is Lipschitz? Show that if $f$ is Lipschitz then it is uniformly continuous. Show also that if $\left(x_{n}\right)_{n}$ is a Cauchy sequence in $X$, and $f$ is uniformly continuous, then the sequence $\left(f\left(x_{n}\right)\right)_{n}$ is convergent.

  (b) Let $f: X \rightarrow \mathbb{R}$ be continuous, and $X$ be sequentially compact. Show that $f$ is uniformly continuous. Is $f$ necessarily Lipschitz? Justify your answer.

  (c) Let $Y$ be a dense subset of $X$, and let $g: Y \rightarrow \mathbb{R}$ be a continuous function. Show that there exists at most one continuous function $f: X \rightarrow \mathbb{R}$ such that for all $y \in Y$, $f(y)=g(y)$. Prove that if $g$ is uniformly continuous, then such a function $f$ exists, and is uniformly continuous.

  [A subset $Y \subset X$ is dense if for any nonempty open subset $U \subset X$, the intersection $U \cap Y$ is nonempty.]

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• Paper 2, Section II, G

  Analysis II

  Part IB, 2016

  (a) What is a norm on a real vector space?

  (b) Let $L\left(\mathbb{R}^{m}, \mathbb{R}^{n}\right)$ be the space of linear maps from $\mathbb{R}^{m}$ to $\mathbb{R}^{n}$. Show that

  $$\|A\|=\sup _{0 \neq x \in \mathbb{R}^{m}} \frac{\|A x\|}{\|x\|}, \quad A \in L\left(\mathbb{R}^{m}, \mathbb{R}^{n}\right),$$

  defines a norm on $L\left(\mathbb{R}^{m}, \mathbb{R}^{n}\right)$, and that if $B \in L\left(\mathbb{R}^{\ell}, \mathbb{R}^{m}\right)$ then $\|A B\| \leqslant\|A\|\|B\|$.

  (c) Let $M_{n}$ be the space of $n \times n$ real matrices, identified with $L\left(\mathbb{R}^{n}, \mathbb{R}^{n}\right)$ in the usual way. Let $U \subset M_{n}$ be the subset

  $$U=\left\{X \in M_{n} \mid I-X \text { is invertible }\right\}$$

  Show that $U$ is an open subset of $M_{n}$ which contains the set $V=\left\{X \in M_{n} \mid\|X\|<1\right\}$.

  (d) Let $f: U \rightarrow M_{n}$ be the map $f(X)=(I-X)^{-1}$. Show carefully that the series $\sum_{k=0}^{\infty} X^{k}$ converges on $V$ to $f(X)$. Hence or otherwise, show that $f$ is twice differentiable at 0 , and compute its first and second derivatives there.

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• Paper 4, Section I, G

  Complex Analysis

  Part IB, 2016

  State carefully Rouché's theorem. Use it to show that the function $z^{4}+3+e^{i z}$ has exactly one zero $z=z_{0}$ in the quadrant

  $$\{z \in \mathbb{C} \mid 0<\arg (z)<\pi / 2\}$$

  and that $\left|z_{0}\right| \leqslant \sqrt{2}$.

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• Paper 3, Section II, G

  Complex Analysis

  Part IB, 2016

  (a) Prove Cauchy's theorem for a triangle.

  (b) Write down an expression for the winding number $I(\gamma, a)$ of a closed, piecewise continuously differentiable curve $\gamma$ about a point $a \in \mathbb{C}$ which does not lie on $\gamma$.

  (c) Let $U \subset \mathbb{C}$ be a domain, and $f: U \rightarrow \mathbb{C}$ a holomorphic function with no zeroes in $U$. Suppose that for infinitely many positive integers $k$ the function $f$ has a holomorphic $k$-th root. Show that there exists a holomorphic function $F: U \rightarrow \mathbb{C}$ such that $f=\exp F$.

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• Paper 2, Section II, A

  Complex Analysis or Complex Methods

  Part IB, 2016

  Let $a=N+1 / 2$ for a positive integer $N$. Let $C_{N}$ be the anticlockwise contour defined by the square with its four vertices at $a \pm i a$ and $-a \pm i a$. Let

  $$I_{N}=\oint_{C_{N}} \frac{d z}{z^{2} \sin (\pi z)}$$

  Show that $1 / \sin (\pi z)$ is uniformly bounded on the contours $C_{N}$ as $N \rightarrow \infty$, and hence that $I_{N} \rightarrow 0$ as $N \rightarrow \infty$.

  Using this result, establish that

  $$\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n^{2}}=\frac{\pi^{2}}{12}$$

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• Paper 1, Section II, A

  Complex Analysis or Complex Methods

  Part IB, 2016

  Let $w=u+i v$ and let $z=x+i y$, for $u, v, x, y$ real.

  (a) Let A be the map defined by $w=\sqrt{z}$, using the principal branch. Show that A maps the region to the left of the parabola $y^{2}=4(1-x)$ on the $z-$ plane, with the negative real axis $x \in(-\infty, 0]$ removed, into the vertical strip of the $w-$ plane between the lines $u=0$ and $u=1$.

  (b) Let $\mathrm{B}$ be the map defined by $w=\tan ^{2}(z / 2)$. Show that $\mathrm{B}$ maps the vertical strip of the $z$-plane between the lines $x=0$ and $x=\pi / 2$ into the region inside the unit circle on the $w$-plane, with the part $u \in(-1,0]$ of the negative real axis removed.

  (c) Using the results of parts (a) and (b), show that the map C, defined by $w=\tan ^{2}(\pi \sqrt{z} / 4)$, maps the region to the left of the parabola $y^{2}=4(1-x)$ on the $z$-plane, including the negative real axis, onto the unit disc on the $w$-plane.

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• Paper 1, Section I, A

  Complex Analysis or Complex Methods

  Part IB, 2016

  Classify the singularities of the following functions at both $z=0$ and at the point at infinity on the extended complex plane:

  $$\begin{aligned} f_{1}(z) &=\frac{e^{z}}{z \sin ^{2} z}, \\ f_{2}(z) &=\frac{1}{z^{2}(1-\cos z)}, \\ f_{3}(z) &=z^{2} \sin (1 / z) \end{aligned}$$

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• Paper 3, Section I, A

  Complex Methods

  Part IB, 2016

  The function $f(x)$ has Fourier transform

  $$\tilde{f}(k)=\int_{-\infty}^{\infty} f(x) e^{-i k x} d x=\frac{-2 k i}{p^{2}+k^{2}},$$

  where $p>0$ is a real constant. Using contour integration, calculate $f(x)$ for $x<0$. [Jordan's lemma and the residue theorem may be used without proof.]

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• Paper 4, Section II, A

  Complex Methods

  Part IB, 2016

  (a) Show that the Laplace transform of the Heaviside step function $H(t-a)$ is

  $$\int_{0}^{\infty} H(t-a) e^{-p t} d t=\frac{e^{-a p}}{p}$$

  for $a>0$.

  (b) Derive an expression for the Laplace transform of the second derivative of a function $f(t)$ in terms of the Laplace transform of $f(t)$ and the properties of $f(t)$ at $t=0$.

  (c) A bar of length $L$ has its end at $x=L$ fixed. The bar is initially at rest and straight. The end at $x=0$ is given a small fixed transverse displacement of magnitude $a$ at $t=0^{+}$. You may assume that the transverse displacement $y(x, t)$ of the bar satisfies the wave equation with some wave speed $c$, and so the tranverse displacement $y(x, t)$ is the solution to the problem:

  $$\begin{array}{cl} \frac{\partial^{2} y}{\partial t^{2}}=c^{2} \frac{\partial^{2} y}{\partial x^{2}} & \text { for } 0<x<L \text { and } t> \\ y(x, 0)=\frac{\partial y}{\partial t}(x, 0)=0 & \text { for } 0<x<L, \\ y(0, t)=a ; y(L, t)=0 & \text { for } t>0 . \end{array}$$

  (i) Show that the Laplace transform $Y(x, p)$ of $y(x, t)$, defined as

  $$Y(x, p)=\int_{0}^{\infty} y(x, t) e^{-p t} d t$$

  is given by

  $$Y(x, p)=\frac{a \sinh \left[\frac{p}{c}(L-x)\right]}{p \sinh \left[\frac{p L}{c}\right]}$$

  (ii) By use of the binomial theorem or otherwise, express $y(x, t)$ as an infinite series.

  (iii) Plot the transverse displacement of the midpoint of the bar $y(L / 2, t)$ against time.

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• Paper 2, Section I, $6 \mathrm{D}$

  Electromagnetism

  Part IB, 2016

  (a) Derive the integral form of Ampère's law from the differential form of Maxwell's equations with a time-independent magnetic field, $\rho=0$ and $\mathbf{E}=\mathbf{0}$.

  (b) Consider two perfectly-conducting concentric thin cylindrical shells of infinite length with axes along the $z$-axis and radii $a$ and $b(a<b)$. Current $I$ flows in the positive $z$-direction in each shell. Use Ampère's law to calculate the magnetic field in the three regions: (i) $r<a$, (ii) $a<r<b$ and (iii) $r>b$, where $r=\sqrt{x^{2}+y^{2}}$.

  (c) If current $I$ now flows in the positive $z$-direction in the inner shell and in the negative $z$-direction in the outer shell, calculate the magnetic field in the same three regions.

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• Paper 4, Section I, D

  Electromagnetism

  Part IB, 2016

  (a) Starting from Maxwell's equations, show that in a vacuum,

  $$\frac{1}{c^{2}} \frac{\partial^{2} \mathbf{E}}{\partial t^{2}}-\nabla^{2} \mathbf{E}=\mathbf{0} \quad \text { and } \quad \boldsymbol{\nabla} \cdot \mathbf{E}=0 \quad \text { where } \quad c=\sqrt{\frac{1}{\epsilon_{0} \mu_{0}}} .$$

  (b) Suppose that $\mathbf{E}=\frac{E_{0}}{\sqrt{2}}(1,1,0) \cos (k z-\omega t)$ where $E_{0}, k$ and $\omega$ are real constants.

  (i) What are the wavevector and the polarisation? How is $\omega$ related to $k$ ?

  (ii) Find the magnetic field $\mathbf{B}$.

  (iii) Compute and interpret the time-averaged value of the Poynting vector, $\mathbf{S}=\frac{1}{\mu_{0}} \mathbf{E} \times \mathbf{B}$.

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• Paper 1, Section II, D

  Electromagnetism

  Part IB, 2016

  (a) From the differential form of Maxwell's equations with $\mathbf{J}=\mathbf{0}, \mathbf{B}=\mathbf{0}$ and a time-independent electric field, derive the integral form of Gauss's law.

  (b) Derive an expression for the electric field $\mathbf{E}$ around an infinitely long line charge lying along the $z$-axis with charge per unit length $\mu$. Find the electrostatic potential $\phi$ up to an arbitrary constant.

  (c) Now consider the line charge with an ideal earthed conductor filling the region $x>d$. State the boundary conditions satisfied by $\phi$ and $\mathbf{E}$ on the surface of the conductor.

  (d) Show that the same boundary conditions at $x=d$ are satisfied if the conductor is replaced by a second line charge at $x=2 d, y=0$ with charge per unit length $-\mu$.

  (e) Hence or otherwise, returning to the setup in (c), calculate the force per unit length acting on the line charge.

  (f) What is the charge per unit area $\sigma(y, z)$ on the surface of the conductor?

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• Paper 3, Section II, D

  Electromagnetism

  Part IB, 2016

  (a) State Faraday's law of induction for a moving circuit in a time-dependent magnetic field and define all the terms that appear.

  (b) Consider a rectangular circuit DEFG in the $z=0$ plane as shown in the diagram below. There are two rails parallel to the $x$-axis for $x>0$ starting at $\mathrm{D}$ at $(x, y)=(0,0)$ and $G$ at $(0, L)$. A battery provides an electromotive force $\mathcal{E}_{0}$ between $D$ and $G$ driving current in a positive sense around DEFG. The circuit is completed with a bar resistor of resistance $R$, length $L$ and mass $m$ that slides without friction on the rails; it connects $E$ at $(s(t), 0)$ and $F$ at $(s(t), L)$. The rest of the circuit has no resistance. The circuit is in a constant uniform magnetic field $B_{0}$ parallel to the $z$-axis.

  [In parts (i)-(iv) you can neglect any magnetic field due to current flow.]

  

  (i) Calculate the current in the bar and indicate its direction on a diagram of the circuit.

  (ii) Find the force acting on the bar.

  (iii) If the initial velocity and position of the bar are respectively $\dot{s}(0)=v_{0}>0$ and $s(0)=s_{0}>0$, calculate $\dot{s}(t)$ and $s(t)$ for $t>0$.

  (iv) If $\mathcal{E}_{0}=0$, find the total energy dissipated in the circuit after $t=0$ and verify that total energy is conserved.

  (v) Describe qualitatively the effect of the magnetic field caused by the induced current flowing in the circuit when $\mathcal{E}_{0}=0$.

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• Paper 2, Section II, D

  Electromagnetism

  Part IB, 2016

  (a) State the covariant form of Maxwell's equations and define all the quantities that appear in these expressions.

  (b) Show that $\mathbf{E} \cdot \mathbf{B}$ is a Lorentz scalar (invariant under Lorentz transformations) and find another Lorentz scalar involving $\mathbf{E}$ and $\mathbf{B}$.

  (c) In some inertial frame $S$ the electric and magnetic fields are respectively $\mathbf{E}=\left(0, E_{y}, E_{z}\right)$ and $\mathbf{B}=\left(0, B_{y}, B_{z}\right)$. Find the electric and magnetic fields, $\mathbf{E}^{\prime}=\left(0, E_{y}^{\prime}, E_{z}^{\prime}\right)$ and $\mathbf{B}^{\prime}=\left(0, B_{y}^{\prime}, B_{z}^{\prime}\right)$, in another inertial frame $S^{\prime}$ that is related to $S$ by the Lorentz transformation,

  $$\Lambda_{\nu}^{\mu}=\left(\begin{array}{cccc} \gamma & -\gamma v / c & 0 & 0 \\ -\gamma v / c & \gamma & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right)$$

  where $v$ is the velocity of $S^{\prime}$ in $S$ and $\gamma=\left(1-v^{2} / c^{2}\right)^{-1 / 2}$.

  (d) Suppose that $\mathbf{E}=E_{0}(0,1,0)$ and $\mathbf{B}=\frac{E_{0}}{c}(0, \cos \theta, \sin \theta)$ where $0 \leqslant \theta \leqslant \pi / 2$, and $E_{0}$ is a real constant. An observer is moving in $S$ with velocity $v$ parallel to the $x$-axis. What must $v$ be for the electric and magnetic fields to appear to the observer to be parallel? Comment on the case $\theta=\pi / 2$.

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• Paper 1, Section I, C

  Fluid Dynamics

  Part IB, 2016

  Consider the flow field in cartesian coordinates $(x, y, z)$ given by

  $$\mathbf{u}=\left(-\frac{A y}{x^{2}+y^{2}}, \frac{A x}{x^{2}+y^{2}}, U(z)\right)$$

  where $A$ is a constant. Let $\mathcal{D}$ denote the whole of $\mathbb{R}^{3}$ excluding the $z$ axis.

  (a) Determine the conditions on $A$ and $U(z)$ for the flow to be both incompressible and irrotational in $\mathcal{D}$.

  (b) Calculate the circulation along any closed curve enclosing the $z$ axis.

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• Paper 2, Section I, C

  Fluid Dynamics

  Part IB, 2016

  A steady, two-dimensional unidirectional flow of a fluid with dynamic viscosity $\mu$ is set up between two plates at $y=0$ and $y=h$. The plate at $y=0$ is stationary while the plate at $y=h$ moves with constant speed $U \mathbf{e}_{x}$. The fluid is also subject to a constant pressure gradient $-G \mathbf{e}_{x}$. You may assume that the fluid velocity $\mathbf{u}$ has the form $\mathbf{u}=u(y) \mathbf{e}_{x}$.

  (a) State the equation satisfied by $u(y)$ and its boundary conditions.

  (b) Calculate $u(y)$.

  (c) Show that the value of $U$ may be chosen to lead to zero viscous shear stress acting on the bottom plate and calculate the resulting flow rate.

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• Paper 1, Section II, $17 \mathrm{C}$

  Fluid Dynamics

  Part IB, 2016

  (a) For a velocity field $\mathbf{u}$, show that $\mathbf{u} \cdot \boldsymbol{\nabla} \mathbf{u}=\boldsymbol{\nabla}\left(\frac{1}{2} \mathbf{u}^{2}\right)-\mathbf{u} \times \boldsymbol{\omega}$, where $\boldsymbol{\omega}$ is the flow vorticity.

  (b) For a scalar field $H(\mathbf{r})$, show that if $\mathbf{u} \cdot \nabla H=0$, then $H$ is constant along the flow streamlines.

  (c) State the Euler equations satisfied by an inviscid fluid of constant density subject to conservative body forces.

  (i) If the flow is irrotational, show that an exact first integral of the Euler equations may be obtained.

  (ii) If the flow is not irrotational, show that although an exact first integral of the Euler equations may not be obtained, a similar quantity is constant along the flow streamlines provided the flow is steady.

  (iii) If the flow is now in a frame rotating with steady angular velocity $\Omega \mathbf{e}_{z}$, establish that a similar quantity is constant along the flow streamlines with an extra term due to the centrifugal force when the flow is steady.

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• Paper 4, Section II, C

  Fluid Dynamics

  Part IB, 2016

  (a) Show that for an incompressible fluid, $\nabla \times \boldsymbol{\omega}=-\nabla^{2} \mathbf{u}$, where $\boldsymbol{\omega}$ is the flow vorticity,

  (b) State the equation of motion for an inviscid flow of constant density in a rotating frame subject to gravity. Show that, on Earth, the local vertical component of the centrifugal force is small compared to gravity. Present a scaling argument to justify the linearisation of the Euler equations for sufficiently large rotation rates, and hence deduce the linearised version of the Euler equations in a rapidly rotating frame.

  (c) Denoting the rotation rate of the frame as $\boldsymbol{\Omega}=\Omega \mathbf{e}_{z}$, show that the linearised Euler equations may be manipulated to obtain an equation for the velocity field $\mathbf{u}$ in the form

  $$\frac{\partial^{2} \nabla^{2} \mathbf{u}}{\partial t^{2}}+4 \Omega^{2} \frac{\partial^{2} \mathbf{u}}{\partial z^{2}}=\mathbf{0}$$

  (d) Assume that there exist solutions of the form $\mathbf{u}=\mathbf{U}_{0} \exp [i(\mathbf{k} \cdot \mathbf{x}-\omega t)]$. Show that $\omega=\pm 2 \Omega \cos \theta$ where the angle $\theta$ is to be determined.

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• Paper 3, Section II, C

  Fluid Dynamics

  Part IB, 2016

  A layer of thickness $h_{1}$ of a fluid of density $\rho_{1}$ is located above a layer of thickness $h_{2}$ of a fluid of density $\rho_{2}>\rho_{1}$. The two-fluid system is bounded by two impenetrable surfaces at $y=h_{1}$ and $y=-h_{2}$ and is assumed to be two-dimensional (i.e. independent of $z$ ). The fluid is subsequently perturbed, and the interface between the two fluids is denoted $y=\eta(x, t)$.

  (a) Assuming irrotational motion in each fluid, state the equations and boundary conditions satisfied by the flow potentials, $\varphi_{1}$ and $\varphi_{2}$.

  (b) The interface is perturbed by small-amplitude waves of the form $\eta=\eta_{0} e^{i(k x-\omega t)}$, with $\eta_{0} k \ll 1$. State the equations and boundary conditions satisfied by the linearised system.

  (c) Calculate the dispersion relation of the waves relating the frequency $\omega$ to the wavenumber $k$.

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• Paper 1, Section I, F

  Geometry

  Part IB, 2016

  (a) Describe the Poincaré disc model $D$ for the hyperbolic plane by giving the appropriate Riemannian metric.

  (b) Let $a \in D$ be some point. Write down an isometry $f: D \rightarrow D$ with $f(a)=0$.

  (c) Using the Poincaré disc model, calculate the distance from 0 to re $e^{i \theta}$ with $0 \leqslant r<1$

  (d) Using the Poincaré disc model, calculate the area of a disc centred at a point $a \in D$ and of hyperbolic radius $\rho>0$.

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• Paper 3, Section I, $5 F$

  Geometry

  Part IB, 2016

  (a) State Euler's formula for a triangulation of a sphere.

  (b) A sphere is decomposed into hexagons and pentagons with precisely three edges at each vertex. Determine the number of pentagons.

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• Paper 3, Section II, F

  Geometry

  Part IB, 2016

  (a) Define the cross-ratio $\left[z_{1}, z_{2}, z_{3}, z_{4}\right]$ of four distinct points $z_{1}, z_{2}, z_{3}, z_{4} \in \mathbb{C} \cup\{\infty\}$. Show that the cross-ratio is invariant under Möbius transformations. Express $\left[z_{2}, z_{1}, z_{3}, z_{4}\right]$ in terms of $\left[z_{1}, z_{2}, z_{3}, z_{4}\right]$.

  (b) Show that $\left[z_{1}, z_{2}, z_{3}, z_{4}\right]$ is real if and only if $z_{1}, z_{2}, z_{3}, z_{4}$ lie on a line or circle in $\mathbb{C} \cup\{\infty\}$.

  (c) Let $z_{1}, z_{2}, z_{3}, z_{4}$ lie on a circle in $\mathbb{C}$, given in anti-clockwise order as depicted.

  

  Show that $\left[z_{1}, z_{2}, z_{3}, z_{4}\right]$ is a negative real number, and that $\left[z_{2}, z_{1}, z_{3}, z_{4}\right]$ is a positive real number greater than 1 . Show that $\left|\left[z_{1}, z_{2}, z_{3}, z_{4}\right]\right|+1=\left|\left[z_{2}, z_{1}, z_{3}, z_{4}\right]\right|$. Use this to deduce Ptolemy's relation on lengths of edges and diagonals of the inscribed 4-gon:

  $$\left|z_{1}-z_{3}\right|\left|z_{2}-z_{4}\right|=\left|z_{1}-z_{2}\right|\left|z_{3}-z_{4}\right|+\left|z_{2}-z_{3}\right|\left|z_{4}-z_{1}\right|$$

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• Paper 2, Section II, F

  Geometry

  Part IB, 2016

  (a) Let $A B C$ be a hyperbolic triangle, with the angle at $A$ at least $\pi / 2$. Show that the side $B C$ has maximal length amongst the three sides of $A B C$.

  [You may use the hyperbolic cosine formula without proof. This states that if $a, b$ and $c$ are the lengths of $B C, A C$, and $A B$ respectively, and $\alpha, \beta$ and $\gamma$ are the angles of the triangle at $A, B$ and $C$ respectively, then

  $$\cosh a=\cosh b \cosh c-\sinh b \sinh c \cos \alpha .]$$

  (b) Given points $z_{1}, z_{2}$ in the hyperbolic plane, let $w$ be any point on the hyperbolic line segment joining $z_{1}$ to $z_{2}$, and let $w^{\prime}$ be any point not on the hyperbolic line passing through $z_{1}, z_{2}, w$. Show that

  $$\rho\left(w^{\prime}, w\right) \leqslant \max \left\{\rho\left(w^{\prime}, z_{1}\right), \rho\left(w^{\prime}, z_{2}\right)\right\}$$

  where $\rho$ denotes hyperbolic distance.

  (c) The diameter of a hyperbolic triangle $\Delta$ is defined to be

  $$\sup \{\rho(P, Q) \mid P, Q \in \Delta\}$$

  Show that the diameter of $\Delta$ is equal to the length of its longest side.

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• Paper 4, Section II, F

  Geometry

  Part IB, 2016

  Let $\alpha(s)=(f(s), g(s))$ be a simple curve in $\mathbb{R}^{2}$ parameterised by arc length with $f(s)>0$ for all $s$, and consider the surface of revolution $S$ in $\mathbb{R}^{3}$ defined by the parameterisation

  $$\sigma(u, v)=(f(u) \cos v, f(u) \sin v, g(u))$$

  (a) Calculate the first and second fundamental forms for $S$. Show that the Gaussian curvature of $S$ is given by

  $$K=-\frac{f^{\prime \prime}(u)}{f(u)}$$

  (b) Now take $f(s)=\cos s+2, g(s)=\sin s, 0 \leqslant s<2 \pi$. What is the integral of the Gaussian curvature over the surface of revolution $S$ determined by $f$ and $g$ ?

  [You may use the Gauss-Bonnet theorem without proof.]

  (c) Now suppose $S$ has constant curvature $K \equiv 1$, and suppose there are two points $P_{1}, P_{2} \in \mathbb{R}^{3}$ such that $S \cup\left\{P_{1}, P_{2}\right\}$ is a smooth closed embedded surface. Show that $S$ is a unit sphere, minus two antipodal points.

  [Do not attempt to integrate an expression of the form $\sqrt{1-C^{2} \sin ^{2} u}$ when $C \neq 1$. Study the behaviour of the surface at the largest and smallest possible values of $u$.]

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• Paper 3, Section I, E

  Groups, Rings and Modules

  Part IB, 2016

  Let $G$ be a group of order $n$. Define what is meant by a permutation representation of $G$. Using such representations, show $G$ is isomorphic to a subgroup of the symmetric group $S_{n}$. Assuming $G$ is non-abelian simple, show $G$ is isomorphic to a subgroup of $A_{n}$. Give an example of a permutation representation of $S_{3}$ whose kernel is $A_{3}$.

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• Paper 4, Section I, $2 \mathrm{E}$

  Groups, Rings and Modules

  Part IB, 2016

  Give the statement and the proof of Eisenstein's criterion. Use this criterion to show $x^{p-1}+x^{p-2}+\cdots+1$ is irreducible in $\mathbb{Q}[x]$ where $p$ is a prime.

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• Paper 2, Section I, E

  Groups, Rings and Modules

  Part IB, 2016

  Let $R$ be an integral domain.

  Define what is meant by the field of fractions $F$ of $R$. [You do not need to prove the existence of $F$.]

  Suppose that $\phi: R \rightarrow K$ is an injective ring homomorphism from $R$ to a field $K$. Show that $\phi$ extends to an injective ring homomorphism $\Phi: F \rightarrow K$.

  Give an example of $R$ and a ring homomorphism $\psi: R \rightarrow S$ from $R$ to a ring $S$ such that $\psi$ does not extend to a ring homomorphism $F \rightarrow S$.

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• Paper 1, Section II, E

  Groups, Rings and Modules

  Part IB, 2016

  (a) Let $I$ be an ideal of a commutative ring $R$ and assume $I \subseteq \bigcup_{i=1}^{n} P_{i}$ where the $P_{i}$ are prime ideals. Show that $I \subseteq P_{i}$ for some $i$.

  (b) Show that $\left(x^{2}+1\right)$ is a maximal ideal of $\mathbb{R}[x]$. Show that the quotient ring $\mathbb{R}[x] /\left(x^{2}+1\right)$ is isomorphic to $\mathbb{C} .$

  (c) For $a, b \in \mathbb{R}$, let $I_{a, b}$ be the ideal $(x-a, y-b)$ in $\mathbb{R}[x, y]$. Show that $I_{a, b}$ is a maximal ideal. Find a maximal ideal $J$ of $\mathbb{R}[x, y]$ such that $J \neq I_{a, b}$ for any $a, b \in \mathbb{R}$. Justify your answers.

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• Paper 3, Section II, E

  Groups, Rings and Modules

  Part IB, 2016

  (a) Define what is meant by an algebraic integer $\alpha$. Show that the ideal

  $$I=\{h \in \mathbb{Z}[x] \mid h(\alpha)=0\}$$

  in $\mathbb{Z}[x]$ is generated by a monic irreducible polynomial $f$. Show that $\mathbb{Z}[\alpha]$, considered as a $\mathbb{Z}$-module, is freely generated by $n$ elements where $n=\operatorname{deg} f$.

  (b) Assume $\alpha \in \mathbb{C}$ satisfies $\alpha^{5}+2 \alpha+2=0$. Is it true that the ideal (5) in $\mathbb{Z}[\alpha]$ is a prime ideal? Is there a ring homomorphism $\mathbb{Z}[\alpha] \rightarrow \mathbb{Z}[\sqrt{-1}]$ ? Justify your answers.

  (c) Show that the only unit elements of $\mathbb{Z}[\sqrt{-5}]$ are 1 and $-1$. Show that $\mathbb{Z}[\sqrt{-5}]$ is not a UFD.

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• Paper 4, Section II, E

  Groups, Rings and Modules

  Part IB, 2016

  Let $R$ be a Noetherian ring and let $M$ be a finitely generated $R$-module.

  (a) Show that every submodule of $M$ is finitely generated.

  (b) Show that each maximal element of the set

  $$\mathcal{A}=\{\operatorname{Ann}(m) \mid 0 \neq m \in M\}$$

  is a prime ideal. [Here, maximal means maximal with respect to inclusion, and $\operatorname{Ann}(m)=\{r \in R \mid r m=0\} .]$

  (c) Show that there is a chain of submodules

  $$0=M_{0} \subseteq M_{1} \subseteq \cdots \subseteq M_{l}=M$$

  such that for each $0<i \leqslant l$ the quotient $M_{i} / M_{i-1}$ is isomorphic to $R / P_{i}$ for some prime ideal $P_{i}$.

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• Paper 2, Section II, E

  Groups, Rings and Modules

  Part IB, 2016

  (a) State Sylow's theorems and give the proof of the second theorem which concerns conjugate subgroups.

  (b) Show that there is no simple group of order 351 .

  (c) Let $k$ be the finite field $\mathbb{Z} /(31)$ and let $G L_{2}(k)$ be the multiplicative group of invertible $2 \times 2$ matrices over $k$. Show that every Sylow 3-subgroup of $G L_{2}(k)$ is abelian.

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• Paper 4, Section I, F

  Linear Algebra

  Part IB, 2016

  For which real numbers $x$ do the vectors

  $$(x, 1,1,1), \quad(1, x, 1,1), \quad(1,1, x, 1), \quad(1,1,1, x),$$

  not form a basis of $\mathbb{R}^{4}$ ? For each such value of $x$, what is the dimension of the subspace of $\mathbb{R}^{4}$ that they span? For each such value of $x$, provide a basis for the spanned subspace, and extend this basis to a basis of $\mathbb{R}^{4}$.

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• Paper 2, Section I, F

  Linear Algebra

  Part IB, 2016

  Find a linear change of coordinates such that the quadratic form

  $$2 x^{2}+8 x y-6 x z+y^{2}-4 y z+2 z^{2}$$

  takes the form

  $$\alpha x^{2}+\beta y^{2}+\gamma z^{2}$$

  for real numbers $\alpha, \beta$ and $\gamma$.

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• Paper 1, Section $I$, $1 F$

  Linear Algebra

  Part IB, 2016

  (a) Consider the linear transformation $\alpha: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}$ given by the matrix

  $$\left(\begin{array}{rrr} 5 & -6 & -6 \\ -1 & 4 & 2 \\ 3 & -6 & -4 \end{array}\right)$$

  Find a basis of $\mathbb{R}^{3}$ in which $\alpha$ is represented by a diagonal matrix.

  (b) Give a list of $6 \times 6$ matrices such that any linear transformation $\beta: \mathbb{R}^{6} \rightarrow \mathbb{R}^{6}$ with characteristic polynomial

  $$(x-2)^{4}(x+7)^{2}$$

  and minimal polynomial

  $$(x-2)^{2}(x+7)$$

  is similar to one of the matrices on your list. No two distinct matrices on your list should be similar. [No proof is required.]

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• Paper 1, Section II, F

  Linear Algebra

  Part IB, 2016

  Let $M_{n, n}$ denote the vector space over $F=\mathbb{R}$ or $\mathbb{C}$ of $n \times n$ matrices with entries in $F$. Let $\operatorname{Tr}: M_{n, n} \rightarrow F$ denote the trace functional, i.e., if $A=\left(a_{i j}\right)_{1 \leqslant i, j \leqslant n} \in M_{n, n}$, then

  $$\operatorname{Tr}(A)=\sum_{i=1}^{n} a_{i i}$$

  (a) Show that Tr is a linear functional.

  (b) Show that $\operatorname{Tr}(A B)=\operatorname{Tr}(B A)$ for $A, B \in M_{n, n}$.

  (c) Show that $\operatorname{Tr}$ is unique in the following sense: If $f: M_{n, n} \rightarrow F$ is a linear functional such that $f(A B)=f(B A)$ for each $A, B \in M_{n, n}$, then $f$ is a scalar multiple of the trace functional. If, in addition, $f(I)=n$, then $f=$ Tr.

  (d) Let $W \subseteq M_{n, n}$ be the subspace spanned by matrices $C$ of the form $C=A B-B A$ for $A, B \in M_{n, n}$. Show that $W$ is the kernel of Tr.

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• Paper 4, Section II, F

  Linear Algebra

  Part IB, 2016

  (a) Let $\alpha: V \rightarrow W$ be a linear transformation between finite dimensional vector spaces over a field $F=\mathbb{R}$ or $\mathbb{C}$.

  Define the dual map of $\alpha$. Let $\delta$ be the dual map of $\alpha$. Given a subspace $U \subseteq V$, define the annihilator $U^{\circ}$ of $U$. Show that $(\operatorname{ker} \alpha)^{\circ}$ and the image of $\delta$ coincide. Conclude that the dimension of the image of $\alpha$ is equal to the dimension of the image of $\delta$. Show that $\operatorname{dim} \operatorname{ker}(\alpha)-\operatorname{dim} \operatorname{ker}(\delta)=\operatorname{dim} V-\operatorname{dim} W$.

  (b) Now suppose in addition that $V, W$ are inner product spaces. Define the adjoint $\alpha^{*}$ of $\alpha$. Let $\beta: U \rightarrow V, \gamma: V \rightarrow W$ be linear transformations between finite dimensional inner product spaces. Suppose that the image of $\beta$ is equal to the kernel of $\gamma$. Then show that $\beta \beta^{*}+\gamma^{*} \gamma$ is an isomorphism.

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• Paper 3, Section II, F

  Linear Algebra

  Part IB, 2016

  Let $\alpha: V \rightarrow V$ be a linear transformation defined on a finite dimensional inner product space $V$ over $\mathbb{C}$. Recall that $\alpha$ is normal if $\alpha$ and its adjoint $\alpha^{*}$ commute. Show that $\alpha$ being normal is equivalent to each of the following statements:

  (i) $\alpha=\alpha_{1}+i \alpha_{2}$ where $\alpha_{1}, \alpha_{2}$ are self-adjoint operators and $\alpha_{1} \alpha_{2}=\alpha_{2} \alpha_{1}$;

  (ii) there is an orthonormal basis for $V$ consisting of eigenvectors of $\alpha$;

  (iii) there is a polynomial $g$ with complex coefficients such that $\alpha^{*}=g(\alpha)$.

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• Paper 2, Section II, F

  Linear Algebra

  Part IB, 2016

  Let $M_{n, n}$ denote the vector space over a field $F=\mathbb{R}$ or $\mathbb{C}$ of $n \times n$ matrices with entries in $F$. Given $B \in M_{n, n}$, consider the two linear transformations $R_{B}, L_{B}: M_{n, n} \rightarrow$ $M_{n, n}$ defined by

  $$L_{B}(A)=B A, \quad R_{B}(A)=A B$$

  (a) Show that $\operatorname{det} L_{B}=(\operatorname{det} B)^{n}$.

  [For parts (b) and (c), you may assume the analogous result $\operatorname{det} R_{B}=(\operatorname{det} B)^{n}$ without proof.]

  (b) Now let $F=\mathbb{C}$. For $B \in M_{n, n}$, write $B^{*}$ for the conjugate transpose of $B$, i.e., $B^{*}:=\bar{B}^{T}$. For $B \in M_{n, n}$, define the linear transformation $M_{B}: M_{n, n} \rightarrow M_{n, n}$ by

  $$M_{B}(A)=B A B^{*}$$

  Show that $\operatorname{det} M_{B}=|\operatorname{det} B|^{2 n}$.

  (c) Again let $F=\mathbb{C}$. Let $W \subseteq M_{n, n}$ be the set of Hermitian matrices. [Note that $W$ is not a vector space over $\mathbb{C}$ but only over $\mathbb{R} .]$ For $B \in M_{n, n}$ and $A \in W$, define $T_{B}(A)=B A B^{*}$. Show that $T_{B}$ is an $\mathbb{R}$-linear operator on $W$, and show that as such,

  $$\operatorname{det} T_{B}=|\operatorname{det} B|^{2 n}$$

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• Paper 4, Section I, H

  Markov Chains

  Part IB, 2016

  Consider two boxes, labelled $\mathrm{A}$ and B. Initially, there are no balls in box $\mathrm{A}$ and $k$ balls in box B. Each minute later, one of the $k$ balls is chosen uniformly at random and is moved to the opposite box. Let $X_{n}$ denote the number of balls in box A at time $n$, so that $X_{0}=0$.

  (a) Find the transition probabilities of the Markov chain $\left(X_{n}\right)_{n \geqslant 0}$ and show that it is reversible in equilibrium.

  (b) Find $\mathbb{E}(T)$, where $T=\inf \left\{n \geqslant 1: X_{n}=0\right\}$ is the next time that all $k$ balls are again in box $B$.

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• Paper 3, Section I, H

  Markov Chains

  Part IB, 2016

  Let $\left(X_{n}\right)_{n \geqslant 0}$ be a Markov chain such that $X_{0}=i$. Prove that

  $$\sum_{n=0}^{\infty} \mathbb{P}_{i}\left(X_{n}=i\right)=\frac{1}{\mathbb{P}_{i}\left(X_{n} \neq i \text { for all } n \geqslant 1\right)}$$

  where $1 / 0=+\infty$. [You may use the strong Markov property without proof.]

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• Paper 2, Section II, H

  Markov Chains

  Part IB, 2016

  (a) Prove that every open communicating class of a Markov chain is transient. Prove that every finite transient communicating class is open. Give an example of a Markov chain with an infinite transient closed communicating class.

  (b) Consider a Markov chain $\left(X_{n}\right)_{n \geqslant 0}$ with state space $\{a, b, c, d\}$ and transition probabilities given by the matrix

  $$P=\left(\begin{array}{cccc} 1 / 3 & 0 & 1 / 3 & 1 / 3 \\ 0 & 1 / 4 & 0 & 3 / 4 \\ 1 / 2 & 1 / 2 & 0 & 0 \\ 0 & 2 / 3 & 0 & 1 / 3 \end{array}\right)$$

  (i) Compute $\mathbb{P}\left(X_{n}=b \mid X_{0}=d\right)$ for a fixed $n \geqslant 0$.

  (ii) Compute $\mathbb{P}\left(X_{n}=c\right.$ for some $\left.n \geqslant 1 \mid X_{0}=a\right)$.

  (iii) Show that $P^{n}$ converges as $n \rightarrow \infty$, and determine the limit.

  [Results from lectures can be used without proof if stated carefully.]

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• Paper 1, Section II, H

  Markov Chains

  Part IB, 2016

  Let $\left(X_{n}\right)_{n \geqslant 0}$ be a simple symmetric random walk on the integers, starting at $X_{0}=0$.

  (a) What does it mean to say that a Markov chain is irreducible? What does it mean to say that an irreducible Markov chain is recurrent? Show that $\left(X_{n}\right)_{n} \geqslant 0$ is irreducible and recurrent.

  [Hint: You may find it helpful to use the limit

  $$\lim _{k \rightarrow \infty} \sqrt{k} 2^{-2 k}\left(\begin{array}{c} 2 k \\ k \end{array}\right)=\sqrt{\pi}$$

  You may also use without proof standard necessary and sufficient conditions for recurrence.]

  (b) What does it mean to say that an irreducible Markov chain is positive recurrent? Determine, with proof, whether $\left(X_{n}\right)_{n \geqslant 0}$ is positive recurrent.

  (c) Let

  $$T=\inf \left\{n \geqslant 1: X_{n}=0\right\}$$

  be the first time the chain returns to the origin. Compute $\mathbb{E}\left[s^{T}\right]$ for a fixed number $0<s<1$.

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• Paper 2, Section I, A

  Methods

  Part IB, 2016

  Use the method of characteristics to find $u(x, y)$ in the first quadrant $x \geqslant 0, y \geqslant 0$, where $u(x, y)$ satisfies

  $$\frac{\partial u}{\partial x}-2 x \frac{\partial u}{\partial y}=\cos x$$

  with boundary data $u(x, 0)=\cos x$.

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• Paper 4, Section I, A

  Methods

  Part IB, 2016

  Consider the function $f(x)$ defined by

  $$f(x)=x^{2}, \text { for }-\pi<x<\pi$$

  Calculate the Fourier series representation for the $2 \pi$-periodic extension of this function. Hence establish that

  $$\frac{\pi^{2}}{6}=\sum_{n=1}^{\infty} \frac{1}{n^{2}}$$

  and that

  $$\frac{\pi^{2}}{12}=\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^{2}}$$

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• Paper 3, Section $\mathbf{I}$, A

  Methods

  Part IB, 2016

  Calculate the Green's function $G(x ; \xi)$ given by the solution to

  $$\frac{d^{2} G}{d x^{2}}=\delta(x-\xi) ; \quad G(0 ; \xi)=\frac{d G}{d x}(1 ; \xi)=0$$

  where $\xi \in(0,1), x \in(0,1)$ and $\delta(x)$ is the Dirac $\delta$-function. Use this Green's function to calculate an explicit solution $y(x)$ to the boundary value problem

  $$\frac{d^{2} y}{d x^{2}}=x e^{-x}$$

  where $x \in(0,1)$, and $y(0)=y^{\prime}(1)=0$.

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• Paper 1, Section II, A

  Methods

  Part IB, 2016

  (a) Consider the general self-adjoint problem for $y(x)$ on $[a, b]$ :

  $$-\frac{d}{d x}\left[p(x) \frac{d}{d x} y\right]+q(x) y=\lambda w(x) y ; \quad y(a)=y(b)=0$$

  where $\lambda$ is the eigenvalue, and $w(x)>0$. Prove that eigenfunctions associated with distinct eigenvalues are orthogonal with respect to a particular inner product which you should define carefully.

  (b) Consider the problem for $y(x)$ given by

  $$x y^{\prime \prime}+3 y^{\prime}+\left(\frac{1+\lambda}{x}\right) y=0 ; \quad y(1)=y(e)=0 .$$

  (i) Recast this problem into self-adjoint form.

  (ii) Calculate the complete set of eigenfunctions and associated eigenvalues for this problem. [Hint: You may find it useful to make the substitution $\left.x=e^{s} .\right]$

  (iii) Verify that the eigenfunctions associated with distinct eigenvalues are indeed orthogonal.

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• Paper 3, Section II, B

  Methods

  Part IB, 2016

  (a) Show that the Fourier transform of $f(x)=e^{-a^{2} x^{2}}$, for $a>0$, is

  $$\tilde{f}(k)=\frac{\sqrt{\pi}}{a} e^{-\frac{k^{2}}{4 a^{2}}},$$

  stating clearly any properties of the Fourier transform that you use.

  [Hint: You may assume that $\int_{0}^{\infty} e^{-t^{2}} d t=\sqrt{\pi} / 2$.]

  (b) Consider now the Cauchy problem for the diffusion equation in one space dimension, i.e. solving for $\theta(x, t)$ satisfying:

  $$\frac{\partial \theta}{\partial t}=D \frac{\partial^{2} \theta}{\partial x^{2}} \quad \text { with } \theta(x, 0)=g(x)$$

  where $D$ is a positive constant and $g(x)$ is specified. Consider the following property of a solution:

  Property P: If the initial data $g(x)$ is positive and it is non-zero only within a bounded region (i.e. there is a constant $\alpha$ such that $\theta(x, 0)=0$ for all $|x|>\alpha)$, then for any $\epsilon>0$ (however small) and $\beta$ (however large) the solution $\theta(\beta, \epsilon)$ can be non-zero, i.e. the solution can become non-zero arbitrarily far away after an arbitrarily short time.

  Does Property P hold for solutions of the diffusion equation? Justify your answer (deriving any expression for the solution $\theta(x, t)$ that you use).

  (c) Consider now the wave equation in one space dimension:

  $$\frac{\partial^{2} u}{\partial t^{2}}=c^{2} \frac{\partial^{2} u}{\partial x^{2}}$$

  with given initial data $u(x, 0)=\phi(x)$ and $\frac{\partial u}{\partial t}(x, 0)=0$ (and $c$ is a constant).

  Does Property $\mathrm{P}$ (with $g(x)$ and $\theta(\beta, \epsilon)$ now replaced by $\phi(x)$ and $u(\beta, \epsilon)$ respectively) hold for solutions of the wave equation? Justify your answer again as above.

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• Paper 2, Section II, A

  Methods

  Part IB, 2016

  Consider a bar of length $\pi$ with free ends, subject to longitudinal vibrations. You may assume that the longitudinal displacement $y(x, t)$ of the bar satisfies the wave equation with some wave speed $c$ :

  $$\frac{\partial^{2} y}{\partial t^{2}}=c^{2} \frac{\partial^{2} y}{\partial x^{2}}$$

  for $x \in(0, \pi)$ and $t>0$ with boundary conditions:

  $$\frac{\partial y}{\partial x}(0, t)=\frac{\partial y}{\partial x}(\pi, t)=0$$

  for $t>0$. The bar is initially at rest so that

  $$\frac{\partial y}{\partial t}(x, 0)=0$$

  for $x \in(0, \pi)$, with a spatially varying initial longitudinal displacement given by

  $$y(x, 0)=b x$$

  for $x \in(0, \pi)$, where $b$ is a real constant.

  (a) Using separation of variables, show that

  $$y(x, t)=\frac{b \pi}{2}-\frac{4 b}{\pi} \sum_{n=1}^{\infty} \frac{\cos [(2 n-1) x] \cos [(2 n-1) c t]}{(2 n-1)^{2}}$$

  (b) Determine a periodic function $P(x)$ such that this solution may be expressed as

  $$y(x, t)=\frac{1}{2}[P(x+c t)+P(x-c t)]$$

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• Paper 4, Section II, B

  Methods

  Part IB, 2016

  Let $\mathcal{D}$ be a 2-dimensional region in $\mathbb{R}^{2}$ with boundary $\partial \mathcal{D}$. In this question you may assume Green's second identity:

  $$\int_{\mathcal{D}}\left(f \nabla^{2} g-g \nabla^{2} f\right) d A=\int_{\partial \mathcal{D}}\left(f \frac{\partial g}{\partial n}-g \frac{\partial f}{\partial n}\right) d l$$

  where $\frac{\partial}{\partial n}$ denotes the outward normal derivative on $\partial \mathcal{D}$, and $f$ and $g$ are suitably regular functions that include the free space Green's function in two dimensions. You may also assume that the free space Green's function for the Laplace equation in two dimensions is given by

  $$G_{0}\left(\boldsymbol{r}, \boldsymbol{r}_{0}\right)=\frac{1}{2 \pi} \log \left|\boldsymbol{r}-\boldsymbol{r}_{0}\right|$$

  (a) State the conditions required on a function $G\left(\boldsymbol{r}, \boldsymbol{r}_{\mathbf{0}}\right)$ for it to be a Dirichlet Green's function for the Laplace operator on $\mathcal{D}$. Suppose that $\nabla^{2} \psi=0$ on $\mathcal{D}$. Show that if $G$ is a Dirichlet Green's function for $\mathcal{D}$ then

  $$\psi\left(\boldsymbol{r}_{\mathbf{0}}\right)=\int_{\partial \mathcal{D}} \psi(\boldsymbol{r}) \frac{\partial}{\partial n} G\left(\boldsymbol{r}, \boldsymbol{r}_{\mathbf{0}}\right) d l \quad \text { for } \boldsymbol{r}_{\mathbf{0}} \in \mathcal{D}$$

  (b) Consider the Laplace equation $\nabla^{2} \phi=0$ in the quarter space

  $$\mathcal{D}=\{(x, y): x \geqslant 0 \text { and } y \geqslant 0\}$$

  with boundary conditions

  $$\phi(x, 0)=e^{-x^{2}}, \phi(0, y)=e^{-y^{2}} \text { and } \phi(x, y) \rightarrow 0 \text { as } \sqrt{x^{2}+y^{2}} \rightarrow \infty$$

  Using the method of images, show that the solution is given by

  $$\phi\left(x_{0}, y_{0}\right)=F\left(x_{0}, y_{0}\right)+F\left(y_{0}, x_{0}\right),$$

  where

  $$F\left(x_{0}, y_{0}\right)=\frac{4 x_{0} y_{0}}{\pi} \int_{0}^{\infty} \frac{t e^{-t^{2}}}{\left[\left(t-x_{0}\right)^{2}+y_{0}^{2}\right]\left[\left(t+x_{0}\right)^{2}+y_{0}^{2}\right]} d t$$

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• Paper 3, Section I, $3 E$

  Metric and Topological Spaces

  Part IB, 2016

  Let $X$ be a topological space and $A \subseteq X$ be a subset. A limit point of $A$ is a point $x \in X$ such that any open neighbourhood $U$ of $x$ intersects $A$. Show that $A$ is closed if and only if it contains all its limit points. Explain what is meant by the interior Int $(A)$ and the closure $\bar{A}$ of $A$. Show that if $A$ is connected, then $\bar{A}$ is connected.

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• Paper 2, Section I, E

  Metric and Topological Spaces

  Part IB, 2016

  Consider $\mathbb{R}$ and $\mathbb{Q}$ with their usual topologies.

  (a) Show that compact subsets of a Hausdorff topological space are closed. Show that compact subsets of $\mathbb{R}$ are closed and bounded.

  (b) Show that there exists a complete metric space $(X, d)$ admitting a surjective continuous map $f: X \rightarrow \mathbb{Q}$.

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• Paper 1, Section II, E

  Metric and Topological Spaces

  Part IB, 2016

  Let $p$ be a prime number. Define what is meant by the $p$-adic metric $d_{p}$ on $\mathbb{Q}$. Show that for $a, b, c \in \mathbb{Q}$ we have

  $$d_{p}(a, b) \leqslant \max \left\{d_{p}(a, c), d_{p}(c, b)\right\}$$

  Show that the sequence $\left(a_{n}\right)$, where $a_{n}=1+p+\cdots+p^{n-1}$, converges to some element in (D.

  For $a \in \mathbb{Q}$ define $|a|_{p}=d_{p}(a, 0)$. Show that if $a, b \in \mathbb{Q}$ and if $|a|_{p} \neq|b|_{p}$, then

  $$|a+b|_{p}=\max \left\{|a|_{p},|b|_{p}\right\} .$$

  Let $a \in \mathbb{Q}$ and let $B(a, \delta)$ be the open ball with centre $a$ and radius $\delta>0$, with respect to the metric $d_{p}$. Show that $B(a, \delta)$ is a closed subset of $\mathbb{Q}$ with respect to the topology induced by $d_{p}$.

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• Paper 4, Section II, E

  Metric and Topological Spaces

  Part IB, 2016

  (a) Let $X$ be a topological space. Define what is meant by a quotient of $X$ and describe the quotient topology on the quotient space. Give an example in which $X$ is Hausdorff but the quotient space is not Hausdorff.

  (b) Let $T^{2}$ be the 2-dimensional torus considered as the quotient $\mathbb{R}^{2} / \mathbb{Z}^{2}$, and let $\pi: \mathbb{R}^{2} \rightarrow T^{2}$ be the quotient map.

  (i) Let $B(u, r)$ be the open ball in $\mathbb{R}^{2}$ with centre $u$ and radius $r<1 / 2$. Show that $U=\pi(B(u, r))$ is an open subset of $T^{2}$ and show that $\pi^{-1}(U)$ has infinitely many connected components. Show each connected component is homeomorphic to $B(u, r)$.

  (ii) Let $\alpha \in \mathbb{R}$ be an irrational number and let $L \subset \mathbb{R}^{2}$ be the line given by the equation $y=\alpha x$. Show that $\pi(L)$ is dense in $T^{2}$ but $\pi(L) \neq T^{2}$.

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• Paper 1, Section I, D

  Numerical Analysis

  Part IB, 2016

  (a) What are real orthogonal polynomials defined with respect to an inner product $\langle\cdot, \cdot\rangle ?$ What does it mean for such polynomials to be monic?

  (b) Real monic orthogonal polynomials, $p_{n}(x)$, of degree $n=0,1,2, \ldots$, are defined with respect to the inner product,

  $$\langle p, q\rangle=\int_{-1}^{1} w(x) p(x) q(x) d x$$

  where $w(x)$ is a positive weight function. Show that such polynomials obey the three-term recurrence relation,

  $$p_{n+1}(x)=\left(x-\alpha_{n}\right) p_{n}(x)-\beta_{n} p_{n-1}(x),$$

  for appropriate $\alpha_{n}$ and $\beta_{n}$ which should be given in terms of inner products.

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• Paper 4, Section I, D

  Numerical Analysis

  Part IB, 2016

  (a) Define the linear stability domain for a numerical method to solve $\mathbf{y}^{\prime}=\mathbf{f}(t, \mathbf{y})$. What is meant by an A-stable method?

  (b) A two-stage Runge-Kutta scheme is given by

  $$\mathbf{k}_{1}=\mathbf{f}\left(t_{n}, \mathbf{y}_{n}\right), \quad \mathbf{k}_{2}=\mathbf{f}\left(t_{n}+\frac{h}{2}, \mathbf{y}_{n}+\frac{h}{2} \mathbf{k}_{1}\right), \quad \mathbf{y}_{n+1}=\mathbf{y}_{n}+h \mathbf{k}_{2},$$

  where $h$ is the step size and $t_{n}=n h$. Show that the order of this scheme is at least two. For this scheme, find the intersection of the linear stability domain with the real axis. Hence show that this method is not A-stable.

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• Paper 1, Section II, D

  Numerical Analysis

  Part IB, 2016

  (a) Consider a method for numerically solving an ordinary differential equation (ODE) for an initial value problem, $\mathbf{y}^{\prime}=\mathbf{f}(t, \mathbf{y})$. What does it mean for a method to converge over $t \in[0, T]$ where $T \in \mathbb{R}$ ? What is the definition of the order of a method?

  (b) A general multistep method for the numerical solution of an ODE is

  $$\sum_{l=0}^{s} \rho_{l} \mathbf{y}_{n+l}=h \sum_{l=0}^{s} \sigma_{l} \mathbf{f}\left(t_{n+l}, \mathbf{y}_{n+l}\right), \quad n=0,1, \ldots$$

  where $s$ is a fixed positive integer. Show that this method is at least of order $p \geqslant 1$ if and only if

  $$\sum_{l=0}^{s} \rho_{l}=0 \quad \text { and } \quad \sum_{l=0}^{s} l^{k} \rho_{l}=k \sum_{l=0}^{s} l^{k-1} \sigma_{l}, \quad k=1, \ldots, p$$

  (c) State the Dahlquist equivalence theorem regarding the convergence of a multistep method.

  (d) Consider the multistep method,

  $$\mathbf{y}_{n+2}+\theta \mathbf{y}_{n+1}+a \mathbf{y}_{n}=h\left[\sigma_{0} \mathbf{f}\left(t_{n}, \mathbf{y}_{n}\right)+\sigma_{1} \mathbf{f}\left(t_{n+1}, \mathbf{y}_{n+1}\right)+\sigma_{2} \mathbf{f}\left(t_{n+2}, \mathbf{y}_{n+2}\right)\right]$$

  Determine the values of $\sigma_{i}$ and $a$ (in terms of the real parameter $\theta$ ) such that the method is at least third order. For what values of $\theta$ does the method converge?

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• Paper 3, Section II, D

  Numerical Analysis

  Part IB, 2016

  (a) Determine real quadratic functions $a(x), b(x), c(x)$ such that the interpolation formula,

  $$f(x) \approx a(x) f(0)+b(x) f(2)+c(x) f(3),$$

  is exact when $f(x)$ is any real polynomial of degree 2 .

  (b) Use this formula to construct approximations for $f(5)$ and $f^{\prime}(1)$ which are exact when $f(x)$ is any real polynomial of degree 2 . Calculate these approximations for $f(x)=x^{3}$ and comment on your answers.

  (c) State the Peano kernel theorem and define the Peano kernel $K(\theta)$. Use this theorem to find the minimum values of the constants $\alpha$ and $\beta$ such that

  $$\left|f(1)-\frac{1}{3}[f(0)+3 f(2)-f(3)]\right| \leqslant \alpha \max _{\xi \in[0,3]}\left|f^{(2)}(\xi)\right|$$

  and

  $$\left|f(1)-\frac{1}{3}[f(0)+3 f(2)-f(3)]\right| \leqslant \beta\left\|f^{(2)}\right\|_{1}$$

  where $f \in C^{2}[0,3]$. Check that these inequalities hold for $f(x)=x^{3}$.

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• Paper 2, Section II, D

  Numerical Analysis

  Part IB, 2016

  (a) Define a Givens rotation $\Omega^{[p, q]} \in \mathbb{R}^{m \times m}$ and show that it is an orthogonal matrix.

  (b) Define a QR factorization of a matrix $A \in \mathbb{R}^{m \times n}$ with $m \geqslant n$. Explain how Givens rotations can be used to find $Q \in \mathbb{R}^{m \times m}$ and $R \in \mathbb{R}^{m \times n}$.

  (c) Let

  $$\mathbf{A}=\left[\begin{array}{ccc} 3 & 1 & 1 \\ 0 & 4 & 1 \\ 0 & 3 & 2 \\ 0 & 0 & 3 / 4 \end{array}\right], \quad \mathbf{b}=\left[\begin{array}{c} 98 / 25 \\ 25 \\ 25 \\ 0 \end{array}\right]$$

  (i) Find a QR factorization of $A$ using Givens rotations.

  (ii) Hence find the vector $\mathbf{x}^{*} \in \mathbb{R}^{3}$ which minimises $\|A \mathbf{x}-\mathbf{b}\|$, where $\|\cdot\|$ is the Euclidean norm. What is $\left\|\mathrm{A} \mathbf{x}^{*}-\mathbf{b}\right\|$ ?

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• Paper 1, Section I, H

  Optimization

  Part IB, 2016

  $$A=\left(\begin{array}{rrr} 5 & -2 & -5 \\ -2 & 3 & 2 \\ -3 & 6 & 2 \\ 4 & -8 & -6 \end{array}\right)$$

  be the payoff of a two-person zero-sum game, where player I (randomly) picks a row to maximise the expected payoff and player II picks a column to minimise the expected payoff. Find each player's optimal strategy and the value of the game.

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• Paper 2, Section I, H

  Optimization

  Part IB, 2016

  Use the simplex algorithm to find the optimal solution to the linear program:

  $$\operatorname{maximise} 3 x+5 y \text { subject to } \begin{aligned} 8 x+3 y+10 z & \leqslant 9, \quad x, y, z \geqslant 0 \\ 5 x+2 y+4 z & \leqslant 8 \\ 2 x+y+3 z & \leqslant 2 \end{aligned}$$

  Write down the dual problem and find its solution.

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• Paper 4, Section II, H

  Optimization

  Part IB, 2016

  (a) What is the maximal flow problem in a network? Explain the Ford-Fulkerson algorithm. Prove that this algorithm terminates if the initial flow is set to zero and all arc capacities are rational numbers.

  (b) Let $A=\left(a_{i, j}\right)_{i, j}$ be an $n \times n$ matrix. We say that $A$ is doubly stochastic if $0 \leqslant a_{i, j} \leqslant 1$ for $i, j$ and

  $$\begin{aligned} &\sum_{i=1}^{n} a_{i, j}=1 \text { for all } j \\ &\sum_{j=1}^{n} a_{i, j}=1 \text { for all } i \end{aligned}$$

  We say that $A$ is a permutation matrix if $a_{i, j} \in\{0,1\}$ for all $i, j$ and

  for all $j$ there exists a unique $i$ such that $a_{i, j}=1$,

  for all $i$ there exists a unique $j$ such that $a_{i, j}=1$.

  Let $\mathcal{C}$ be the set of all $n \times n$ doubly stochastic matrices. Show that a matrix $A$ is an extreme point of $\mathcal{C}$ if and only if $A$ is a permutation matrix.

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• Paper 3, Section II, H

  Optimization

  Part IB, 2016

  (a) State and prove the Lagrangian sufficiency theorem.

  (b) Let $n \geqslant 1$ be a given constant, and consider the problem:

  $$\operatorname{minimise} \sum_{i=1}^{n}\left(2 y_{i}^{2}+x_{i}^{2}\right) \text { subject to } x_{i}=1+\sum_{k=1}^{i} y_{k} \text { for all } i=1, \ldots, n$$

  Find, with proof, constants $a, b, A, B$ such that the optimal solution is given by

  $$x_{i}=a 2^{i}+b 2^{-i} \text { and } y_{i}=A 2^{i}+B 2^{-i} \text {, for all } i=1, \ldots, n .$$

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• Paper 4, Section I, B

  Quantum Mechanics

  Part IB, 2016

  (a) Define the quantum orbital angular momentum operator $\hat{\boldsymbol{L}}=\left(\hat{L}_{1}, \hat{L}_{2}, \hat{L}_{3}\right)$ in three dimensions, in terms of the position and momentum operators.

  (b) Show that $\left[\hat{L}_{1}, \hat{L}_{2}\right]=i \hbar \hat{L}_{3}$. [You may assume that the position and momentum operators satisfy the canonical commutation relations.]

  (c) Let $\hat{L}^{2}=\hat{L}_{1}^{2}+\hat{L}_{2}^{2}+\hat{L}_{3}^{2}$. Show that $\hat{L}_{1}$ commutes with $\hat{L}^{2}$.

  [In this part of the question you may additionally assume without proof the permuted relations $\left[\hat{L}_{2}, \hat{L}_{3}\right]=i \hbar \hat{L}_{1}$ and $\left.\left[\hat{L}_{3}, \hat{L}_{1}\right]=i \hbar \hat{L}_{2} .\right]$

  [Hint: It may be useful to consider the expression $[\hat{A}, \hat{B}] \hat{B}+\hat{B}[\hat{A}, \hat{B}]$ for suitable operators $\hat{A}$ and $\hat{B}$.]

  (d) Suppose that $\psi_{1}(x, y, z)$ and $\psi_{2}(x, y, z)$ are normalised eigenstates of $\hat{L}_{1}$ with eigenvalues $\hbar$ and $-\hbar$ respectively. Consider the wavefunction

  $$\psi=\frac{1}{2} \psi_{1} \cos \omega t+\frac{\sqrt{3}}{2} \psi_{2} \sin \omega t$$

  with $\omega$ being a positive constant. Find the earliest time $t_{0}>0$ such that the expectation value of $\hat{L}_{1}$ in $\psi$ is zero.

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• Paper 3, Section I, B

  Quantum Mechanics

  Part IB, 2016

  (a) Consider a quantum particle moving in one space dimension, in a timeindependent real potential $V(x)$. For a wavefunction $\psi(x, t)$, define the probability density $\rho(x, t)$ and probability current $j(x, t)$ and show that

  $$\frac{\partial \rho}{\partial t}+\frac{\partial j}{\partial x}=0$$

  (b) Suppose now that $V(x)=0$ and $\psi(x, t)=\left(e^{i k x}+R e^{-i k x}\right) e^{-i E t / \hbar}$, where $E=\hbar^{2} k^{2} /(2 m), k$ and $m$ are real positive constants, and $R$ is a complex constant. Compute the probability current for this wavefunction. Interpret the terms in $\psi$ and comment on how this relates to the computed expression for the probability current.

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• Paper 1, Section II, B

  Quantum Mechanics

  Part IB, 2016

  (a) A particle of mass $m$ in one space dimension is confined to move in a potential $V(x)$ given by

  $$V(x)= \begin{cases}0 & \text { for } 0<x<a \\ \infty & \text { for } x<0 \text { or } x>a\end{cases}$$

  The normalised initial wavefunction of the particle at time $t=0$ is

  $$\psi_{0}(x)=\frac{4}{\sqrt{5 a}} \sin ^{3}\left(\frac{\pi x}{a}\right)$$

  (i) Find the expectation value of the energy at time $t=0$.

  (ii) Find the wavefunction of the particle at time $t=1$.

  [Hint: It may be useful to recall the identity $\sin 3 \theta=3 \sin \theta-4 \sin ^{3} \theta$.]

  (b) The right hand wall of the potential is lowered to a finite constant value $U_{0}>0$ giving the new potential:

  $$U(x)= \begin{cases}0 & \text { for } 0<x<a \\ \infty & \text { for } x<0 \\ U_{0} & \text { for } x>a\end{cases}$$

  This potential is set up in the laboratory but the value of $U_{0}$ is unknown. The stationary states of the potential are investigated and it is found that there exists exactly one bound state. Show that the value of $U_{0}$ must satisfy

  $$\frac{\pi^{2} \hbar^{2}}{8 m a^{2}}<U_{0}<\frac{9 \pi^{2} \hbar^{2}}{8 m a^{2}}$$

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• Paper 3, Section II, B

  Quantum Mechanics

  Part IB, 2016

  The spherically symmetric bound state wavefunctions $\psi(r)$ for the Coulomb potential $V=-e^{2} /\left(4 \pi \epsilon_{0} r\right)$ are normalisable solutions of the equation

  $$\frac{d^{2} \psi}{d r^{2}}+\frac{2}{r} \frac{d \psi}{d r}+\frac{2 \lambda}{r} \psi=-\frac{2 m E}{\hbar^{2}} \psi$$

  Here $\lambda=\left(m e^{2}\right) /\left(4 \pi \epsilon_{0} \hbar^{2}\right)$ and $E<0$ is the energy of the state.

  (a) By writing the wavefunction as $\psi(r)=f(r) \exp (-K r)$, for a suitable constant $K$ that you should determine, show that there are normalisable wavefunctions $\psi(r)$ only for energies of the form

  $$E=\frac{-m e^{4}}{32 \pi^{2} \epsilon_{0}^{2} \hbar^{2} N^{2}}$$

  with $N$ being a positive integer.

  (b) The energies in (a) reproduce the predictions of the Bohr model of the hydrogen atom. How do the wavefunctions above compare to the assumptions in the Bohr model?

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• Paper 2, Section II, B

  Quantum Mechanics

  Part IB, 2016

  The one dimensional quantum harmonic oscillator has Hamiltonian

  $$\hat{H}=\frac{1}{2 m} \hat{p}^{2}+\frac{1}{2} m \omega^{2} \hat{x}^{2},$$

  where $m$ and $\omega$ are real positive constants and $\hat{x}$ and $\hat{p}$ are the standard position and momentum operators satisfying the commutation relation $[\hat{x}, \hat{p}]=i \hbar$. Consider the operators

  $$\hat{A}=\hat{p}-i m \omega \hat{x} \quad \text { and } \quad \hat{B}=\hat{p}+i m \omega \hat{x} .$$

  (a) Show that

  $$\hat{B} \hat{A}=2 m\left(\hat{H}-\frac{1}{2} \hbar \omega\right) \quad \text { and } \quad \hat{A} \hat{B}=2 m\left(\hat{H}+\frac{1}{2} \hbar \omega\right) .$$

  (b) Suppose that $\phi$ is an eigenfunction of $\hat{H}$ with eigenvalue $E$. Show that $\hat{A} \phi$ is then also an eigenfunction of $\hat{H}$ and that its corresponding eigenvalue is $E-\hbar \omega$.

  (c) Show that for any normalisable wavefunctions $\chi$ and $\psi$,

  $$\int_{-\infty}^{\infty} \chi^{*}(\hat{A} \psi) d x=\int_{-\infty}^{\infty}(\hat{B} \chi)^{*} \psi d x$$

  [You may assume that the operators $\hat{x}$ and $\hat{p}$ are Hermitian.]

  (d) With $\phi$ as in (b), obtain an expression for the norm of $\hat{A} \phi$ in terms of $E$ and the norm of $\phi$. [The squared norm of any wavefunction $\psi$ is $\int_{-\infty}^{\infty}|\psi|^{2} d x$.]

  (e) Show that all eigenvalues of $\hat{H}$ are non-negative.

  (f) Using the above results, deduce that each eigenvalue $E$ of $\hat{H}$ must be of the form $E=\left(n+\frac{1}{2}\right) \hbar \omega$ for some non-negative integer $n$.

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• Paper 1, Section I, H

  Statistics

  Part IB, 2016

  Let $X_{1}, \ldots, X_{n}$ be independent samples from the exponential distribution with density $f(x ; \lambda)=\lambda e^{-\lambda x}$ for $x>0$, where $\lambda$ is an unknown parameter. Find the critical region of the most powerful test of size $\alpha$ for the hypotheses $H_{0}: \lambda=1$ versus $H_{1}: \lambda=2$. Determine whether or not this test is uniformly most powerful for testing $H_{0}^{\prime}: \lambda \leqslant 1$ versus $H_{1}^{\prime}: \lambda>1$.

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• Paper 2, Section I, H

  Statistics

  Part IB, 2016

  The efficacy of a new medicine was tested as follows. Fifty patients were given the medicine, and another fifty patients were given a placebo. A week later, the number of patients who got better, stayed the same, or got worse was recorded, as summarised in this table:

  \begin{tabular}{|l|c|c|} \hline & medicine & placebo \ better & 28 & 22 \ same & 4 & 16 \ worse & 18 & 12 \ \hline \end{tabular}

  Conduct a Pearson chi-squared test of size $1 \%$ of the hypothesis that the medicine and the placebo have the same effect.

  [Hint: You may find the following values relevant:

  $\left.\begin{array}{lcccccc}\text { Distribution } & \chi_{1}^{2} & \chi_{2}^{2} & \chi_{3}^{2} & \chi_{4}^{2} & \chi_{5}^{2} & \chi_{6}^{2} \\ 99 \% \text { percentile } & 6.63 & 9.21 & 11.34 & 13.3 & 15.09 & 16.81 .\end{array}\right]$

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• Paper 4, Section II, H

  Statistics

  Part IB, 2016

  Consider the linear regression model

  $$Y_{i}=\alpha+\beta x_{i}+\varepsilon_{i}$$

  for $i=1, \ldots, n$, where the non-zero numbers $x_{1}, \ldots, x_{n}$ are known and are such that $x_{1}+\ldots+x_{n}=0$, the independent random variables $\varepsilon_{1}, \ldots, \varepsilon_{n}$ have the $N\left(0, \sigma^{2}\right)$ distribution, and the parameters $\alpha, \beta$ and $\sigma^{2}$ are unknown.

  (a) Let $(\hat{\alpha}, \hat{\beta})$ be the maximum likelihood estimator of $(\alpha, \beta)$. Prove that for each $i$, the random variables $\hat{\alpha}, \hat{\beta}$ and $Y_{i}-\hat{\alpha}-\hat{\beta} x_{i}$ are uncorrelated. Using standard facts about the multivariate normal distribution, prove that $\hat{\alpha}, \hat{\beta}$ and $\sum_{i=1}^{n}\left(Y_{i}-\hat{\alpha}-\hat{\beta} x_{i}\right)^{2}$ are independent.

  (b) Find the critical region of the generalised likelihood ratio test of size $5 \%$ for testing $H_{0}: \alpha=0$ versus $H_{1}: \alpha \neq 0$. Prove that the power function of this test is of the form $w\left(\alpha, \beta, \sigma^{2}\right)=g(\alpha / \sigma)$ for some function $g$. [You are not required to find $g$ explicitly.]

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• Paper 1, Section II, H

  Statistics

  Part IB, 2016

  (a) What does it mean to say a statistic $T$ is sufficient for an unknown parameter $\theta$ ? State the factorisation criterion for sufficiency and prove it in the discrete case.

  (b) State and prove the Rao-Blackwell theorem.

  (c) Let $X_{1}, \ldots, X_{n}$ be independent samples from the uniform distribution on $[-\theta, \theta]$ for an unknown positive parameter $\theta$. Consider the two-dimensional statistic

  $$T=\left(\min _{i} X_{i}, \max _{i} X_{i}\right) .$$

  Prove that $T$ is sufficient for $\theta$. Determine, with proof, whether or not $T$ is minimally sufficient.

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• Paper 3, Section II, H

  Statistics

  Part IB, 2016

  Let $X_{1}, \ldots, X_{n}$ be independent samples from the Poisson distribution with mean $\theta$.

  (a) Compute the maximum likelihood estimator of $\theta$. Is this estimator biased?

  (b) Under the assumption that $n$ is very large, use the central limit theorem to find an approximate $95 \%$ confidence interval for $\theta$. [You may use the notation $z_{\alpha}$ for the number such that $\mathbb{P}\left(Z \geqslant z_{\alpha}\right)=\alpha$ for a standard normal $\left.Z \sim N(0,1) .\right]$

  (c) Now suppose the parameter $\theta$ has the $\Gamma(k, \lambda)$ prior distribution. What is the posterior distribution? What is the Bayes point estimator for $\theta$ for the quadratic loss function $L(\theta, a)=(\theta-a)^{2} ?$ Let $X_{n+1}$ be another independent sample from the same distribution. Given $X_{1}, \ldots, X_{n}$, what is the posterior probability that $X_{n+1}=0$ ?

  [Hint: The density of the $\Gamma(k, \lambda)$ distribution is $f(x ; k, \lambda)=\lambda^{k} x^{k-1} e^{-\lambda x} / \Gamma(k)$, for $x>0$.]

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• Paper 1, Section I, C

  Variational Principles

  Part IB, 2016

  (a) Consider the function $f\left(x_{1}, x_{2}\right)=2 x_{1}^{2}+x_{2}^{2}+\alpha x_{1} x_{2}$, where $\alpha$ is a real constant. For what values of $\alpha$ is the function $f$ convex?

  (b) In the case $\alpha=-3$, calculate the extremum of $x_{1}^{2}$ on the set of points where $f\left(x_{1}, x_{2}\right)+1=0 .$

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• Paper 3, Section I, C

  Variational Principles

  Part IB, 2016

  Two points $A$ and $B$ are located on the curved surface of the circular cylinder of radius $R$ with axis along the $z$-axis. We denote their locations by $\left(R, \phi_{A}, z_{A}\right)$ and $\left(R, \phi_{B}, z_{B}\right)$ using cylindrical polar coordinates and assume $\phi_{A} \neq \phi_{B}, z_{A} \neq z_{B}$. A path $\phi(z)$ is drawn on the cylinder to join $A$ and $B$. Show that the path of minimum distance between the points $A$ and $B$ is a helix, and determine its pitch. [For a helix with axis parallel to the $z$ axis, the pitch is the change in $z$ after one complete helical turn.]

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• Paper 2, Section II, C

  Variational Principles

  Part IB, 2016

  A flexible wire filament is described by the curve $(x, y(x), z(x))$ in cartesian coordinates for $0 \leqslant x \leqslant L$. The filament is assumed to be almost straight and thus we assume $\left|y^{\prime}\right| \ll 1$ and $\left|z^{\prime}\right| \ll 1$ everywhere.

  (a) Show that the total length of the filament is approximately $L+\Delta$ where

  $$\Delta=\frac{1}{2} \int_{0}^{L}\left[\left(y^{\prime}\right)^{2}+\left(z^{\prime}\right)^{2}\right] d x$$

  (b) Under a uniform external axial force, $F>0$, the filament adopts the shape which minimises the total energy, $\mathcal{E}=E_{B}-F \Delta$, where $E_{B}$ is the bending energy given by

  $$E_{B}[y, z]=\frac{1}{2} \int_{0}^{L}\left[A(x)\left(y^{\prime \prime}\right)^{2}+B(x)\left(z^{\prime \prime}\right)^{2}\right] d x$$

  and where $A(x)$ and $B(x)$ are $x$-dependent bending rigidities (both known and strictly positive). The filament satisfies the boundary conditions

  $$y(0)=y^{\prime}(0)=z(0)=z^{\prime}(0)=0, \quad y(L)=y^{\prime}(L)=z(L)=z^{\prime}(L)=0$$

  Derive the Euler-Lagrange equations for $y(x)$ and $z(x)$.

  (c) In the case where $A=B=1$ and $L=1$, show that below a critical force, $F_{c}$, which should be determined, the only energy-minimising solution for the filament is straight $(y=z=0)$, but that a new nonzero solution is admissible at $F=F_{c}$.

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• Paper 4, Section II, C

  Variational Principles

  Part IB, 2016

  A fish swims in the ocean along a straight line with speed $V(t)$. The fish starts its journey from rest (zero velocity at $t=0$ ) and, during a given time $T$, swims subject to the constraint that the total distance travelled is $L$. The energy cost for swimming is $a V^{2}+b \dot{V}^{2}$ per unit time, where $a, b \geqslant 0$ are known and $a^{2}+b^{2} \neq 0$.

  (a) Derive the Euler-Lagrange condition on $V(t)$ for the journey to have minimum energetic cost.

  (b) In the case $a \neq 0, b \neq 0$ solve for $V(t)$ assuming that the fish starts at $t=0$ with zero acceleration (in addition to zero velocity).

  (c) In the case $a=0$, the fish can decide between three different boundary conditions for its journey. In addition to starting with zero velocity, it can:

  (1) start at $t=0$ with zero acceleration;

  (2) end at $t=T$ with zero velocity; or

  (3) end at $t=T$ with zero acceleration.

  Which of $(1),(2)$ or (3) is the best minimal-energy cost strategy?

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• Paper 3, Section II, H

  Algebraic Geometry

  Part II, 2016

  (a) Let $X$ be an affine variety. Define the tangent space of $X$ at a point $P$. Say what it means for the variety to be singular at $P$.

  Define the dimension of $X$ in terms of (i) the tangent spaces of $X$, and (ii) Krull dimension.

  (b) Consider the ideal $I$ generated by the set $\left\{y, y^{2}-x^{3}+x y^{3}\right\} \subseteq k[x, y]$. What is $Z(I) \subseteq \mathbb{A}^{2} ?$

  Using the generators of the ideal, calculate the tangent space of a point in $Z(I)$. What has gone wrong? [A complete argument is not necessary.]

  (c) Calculate the dimension of the tangent space at each point $p \in X$ for $X=$ $Z\left(x-y^{2}, x-z w\right) \subseteq \mathbb{A}^{4}$, and determine the location of the singularities of $X .$

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• Paper 2, Section II, H

  Algebraic Geometry

  Part II, 2016

  In this question we work over an algebraically closed field of characteristic zero. Let $X^{o}=Z\left(x^{6}+x y^{5}+y^{6}-1\right) \subset \mathbb{A}^{2}$ and let $X \subset \mathbb{P}^{2}$ be the closure of $X^{o}$ in $\mathbb{P}^{2} .$

  (a) Show that $X$ is a non-singular curve.

  (b) Show that $\omega=d x /\left(5 x y^{4}+6 y^{5}\right)$ is a regular differential on $X$.

  (c) Compute the divisor of $\omega$. What is the genus of $X$ ?

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• Paper 4, Section II, H

  Algebraic Geometry

  Part II, 2016

  (a) Let $C$ be a smooth projective curve, and let $D$ be an effective divisor on $C$. Explain how $D$ defines a morphism $\phi_{D}$ from $C$ to some projective space.

  State a necessary and sufficient condition on $D$ so that the pull-back of a hyperplane via $\phi_{D}$ is an element of the linear system $|D|$.

  State necessary and sufficient conditions for $\phi_{D}$ to be an isomorphism onto its image.

  (b) Let $C$ now have genus 2 , and let $K$ be an effective canonical divisor. Show that the morphism $\phi_{K}$ is a morphism of degree 2 from $C$ to $\mathbb{P}^{1}$.

  Consider the divisor $K+P_{1}+P_{2}$ for points $P_{i}$ with $P_{1}+P_{2} \nsim K$. Show that the linear system associated to this divisor induces a morphism $\phi$ from $C$ to a quartic curve in $\mathbb{P}^{2}$. Show furthermore that $\phi(P)=\phi(Q)$, with $P \neq Q$, if and only if $\{P, Q\}=\left\{P_{1}, P_{2}\right\}$.

  [You may assume the Riemann-Roch theorem.]

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• Paper 1, Section II, H

  Algebraic Geometry

  Part II, 2016

  Let $k$ be an algebraically closed field.

  (a) Let $X$ and $Y$ be affine varieties defined over $k$. Given a map $f: X \rightarrow Y$, define what it means for $f$ to be a morphism of affine varieties.

  (b) Let $f: \mathbb{A}^{1} \rightarrow \mathbb{A}^{3}$ be the map given by

  $$f(t)=\left(t, t^{2}, t^{3}\right)$$

  Show that $f$ is a morphism. Show that the image of $f$ is a closed subvariety of $\mathbb{A}^{3}$ and determine its ideal.

  (c) Let $g: \mathbb{P}^{1} \times \mathbb{P}^{1} \times \mathbb{P}^{1} \rightarrow \mathbb{P}^{7}$ be the map given by

  $g\left(\left(s_{1}, t_{1}\right),\left(s_{2}, t_{2}\right),\left(s_{3}, t_{3}\right)\right)=\left(s_{1} s_{2} s_{3}, s_{1} s_{2} t_{3}, s_{1} t_{2} s_{3}, s_{1} t_{2} t_{3}, t_{1} s_{2} s_{3}, t_{1} s_{2} t_{3}, t_{1} t_{2} s_{3}, t_{1} t_{2} t_{3}\right) .$

  Show that the image of $g$ is a closed subvariety of $\mathbb{P}^{7}$.

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• Paper 3, Section II, G

  Algebraic Topology

  Part II, 2016

  Construct a space $X$ as follows. Let $Z_{1}, Z_{2}, Z_{3}$ each be homeomorphic to the standard 2-sphere $S^{2} \subseteq \mathbb{R}^{3}$. For each $i$, let $x_{i} \in Z_{i}$ be the North pole $(1,0,0)$ and let $y_{i} \in Z_{i}$ be the South pole $(-1,0,0)$. Then

  $$X=\left(Z_{1} \sqcup Z_{2} \sqcup Z_{3}\right) / \sim$$

  where $x_{i+1} \sim y_{i}$ for each $i$ (and indices are taken modulo 3 ).

  (a) Describe the universal cover of $X$.

  (b) Compute the fundamental group of $X$ (giving your answer as a well-known group).

  (c) Show that $X$ is not homotopy equivalent to the circle $S^{1}$.

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• Paper 2, Section II, G

  Algebraic Topology

  Part II, 2016

  (a) Let $K, L$ be simplicial complexes, and $f:|K| \rightarrow|L|$ a continuous map. What does it mean to say that $g: K \rightarrow L$ is a simplicial approximation to $f ?$

  (b) Define the barycentric subdivision of a simplicial complex $K$, and state the Simplicial Approximation Theorem.

  (c) Show that if $g$ is a simplicial approximation to $f$ then $f \simeq|g|$.

  (d) Show that the natural inclusion $\left|K^{(1)}\right| \rightarrow|K|$ induces a surjective map on fundamental groups.

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• Paper 1, Section II, G

  Algebraic Topology

  Part II, 2016

  Let $T=S^{1} \times S^{1}$ be the 2-dimensional torus. Let $\alpha: S^{1} \rightarrow T$ be the inclusion of the coordinate circle $S^{1} \times\{1\}$, and let $X$ be the result of attaching a 2-cell along $\alpha$.

  (a) Write down a presentation for the fundamental group of $X$ (with respect to some basepoint), and identify it with a well-known group.

  (b) Compute the simplicial homology of any triangulation of $X$.

  (c) Show that $X$ is not homotopy equivalent to any compact surface.

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• Paper 4, Section II, G

  Algebraic Topology

  Part II, 2016

  Let $T=S^{1} \times S^{1}$ be the 2-dimensional torus, and let $X$ be constructed from $T$ by removing a small open disc.

  (a) Show that $X$ is homotopy equivalent to $S^{1} \vee S^{1}$.

  (b) Show that the universal cover of $X$ is homotopy equivalent to a tree.

  (c) Exhibit (finite) cell complexes $X, Y$, such that $X$ and $Y$ are not homotopy equivalent but their universal covers $\widetilde{X}, \widetilde{Y}$ are.

  [State carefully any results from the course that you use.]

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• Paper 1, Section II, A

  Applications of Quantum Mechanics

  Part II, 2016

  A particle in one dimension of mass $m$ and energy $E=\hbar^{2} k^{2} / 2 m(k>0)$ is incident from $x=-\infty$ on a potential $V(x)$ with $V(x) \rightarrow 0$ as $x \rightarrow-\infty$ and $V(x)=\infty$ for $x>0$. The relevant solution of the time-independent Schrödinger equation has the asymptotic form

  $$\psi(x) \sim \exp (i k x)+r(k) \exp (-i k x), \quad x \rightarrow-\infty$$

  Explain briefly why a pole in the reflection amplitude $r(k)$ at $k=i \kappa$ with $\kappa>0$ corresponds to the existence of a stable bound state in this potential. Indicate why a pole in $r(k)$ just below the real $k$-axis, at $k=k_{0}-i \rho$ with $k_{0} \gg \rho>0$, corresponds to a quasi-stable bound state. Find an approximate expression for the lifetime $\tau$ of such a quasi-stable state.

  Now suppose that

  $$V(x)= \begin{cases}\left(\hbar^{2} U / 2 m\right) \delta(x+a) & \text { for } x<0 \\ \infty & \text { for } x>0\end{cases}$$

  where $U>0$ and $a>0$ are constants. Compute the reflection amplitude $r(k)$ in this case and deduce that there are quasi-stable bound states if $U$ is large. Give expressions for the wavefunctions and energies of these states and compute their lifetimes, working to leading non-vanishing order in $1 / U$ for each expression.

  [ You may assume $\psi=0$ for $x \geqslant 0$ and $\lim _{\epsilon \rightarrow 0+}\left\{\psi^{\prime}(-a+\epsilon)-\psi^{\prime}(-a-\epsilon)\right\}=U \psi(-a)$.]

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• Paper 4, Section II, A

  Applications of Quantum Mechanics

  Part II, 2016

  Let $\Lambda \subset \mathbb{R}^{2}$ be a Bravais lattice. Define the dual lattice $\Lambda^{*}$ and show that

  $$V(\mathbf{x})=\sum_{\mathbf{q} \in \Lambda^{*}} V_{\mathbf{q}} \exp (i \mathbf{q} \cdot \mathbf{x})$$

  obeys $V(\mathbf{x}+l)=V(\mathbf{x})$ for all $l \in \Lambda$, where $V_{\mathbf{q}}$ are constants. Suppose $V(\mathbf{x})$ is the potential for a particle of mass $m$ moving in a two-dimensional crystal that contains a very large number of lattice sites of $\Lambda$ and occupies an area $\mathcal{A}$. Adopting periodic boundary conditions, plane-wave states $|\mathbf{k}\rangle$ can be chosen such that

  $$\langle\mathbf{x} \mid \mathbf{k}\rangle=\frac{1}{\mathcal{A}^{1 / 2}} \exp (i \mathbf{k} \cdot \mathbf{x}) \quad \text { and } \quad\left\langle\mathbf{k} \mid \mathbf{k}^{\prime}\right\rangle=\delta_{\mathbf{k} \mathbf{k}^{\prime}}$$

  The allowed wavevectors $\mathbf{k}$ are closely spaced and include all vectors in $\Lambda^{*}$. Find an expression for the matrix element $\left\langle\mathbf{k}|V(\mathbf{x})| \mathbf{k}^{\prime}\right\rangle$ in terms of the coefficients $V_{\mathbf{q}}$. [You need not discuss additional details of the boundary conditions.]

  Now suppose that $V(\mathbf{x})=\lambda U(\mathbf{x})$, where $\lambda \ll 1$ is a dimensionless constant. Find the energy $E(\mathbf{k})$ for a particle with wavevector $\mathbf{k}$ to order $\lambda^{2}$ in non-degenerate perturbation theory. Show that this expansion in $\lambda$ breaks down on the Bragg lines in k-space defined by the condition

  $$\mathbf{k} \cdot \mathbf{q}=\frac{1}{2}|\mathbf{q}|^{2} \quad \text { for } \quad \mathbf{q} \in \Lambda^{*}$$

  and explain briefly, without additional calculations, the significance of this for energy levels in the crystal.

  Consider the particular case in which $\Lambda$ has primitive vectors

  $$\mathbf{a}_{1}=2 \pi\left(\mathbf{i}+\frac{1}{\sqrt{3}} \mathbf{j}\right), \quad \mathbf{a}_{2}=2 \pi \frac{2}{\sqrt{3}} \mathbf{j}$$

  where $\mathbf{i}$ and $\mathbf{j}$ are orthogonal unit vectors. Determine the polygonal region in $\mathbf{k}$-space corresponding to the lowest allowed energy band.

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• Paper 2, Section II, A

  Applications of Quantum Mechanics

  Part II, 2016

  A particle of mass $m$ moves in three dimensions subject to a potential $V(\mathbf{r})$ localised near the origin. The wavefunction for a scattering process with incident particle of wavevector $\mathbf{k}$ is denoted $\psi(\mathbf{k}, \mathbf{r})$. With reference to the asymptotic form of $\psi$, define the scattering amplitude $f\left(\mathbf{k}, \mathbf{k}^{\prime}\right)$, where $\mathbf{k}^{\prime}$ is the wavevector of the outgoing particle with $\left|\mathbf{k}^{\prime}\right|=|\mathbf{k}|=k$.

  By recasting the Schrödinger equation for $\psi(\mathbf{k}, \mathbf{r})$ as an integral equation, show that

  $$f\left(\mathbf{k}, \mathbf{k}^{\prime}\right)=-\frac{m}{2 \pi \hbar^{2}} \int d^{3} \mathbf{r}^{\prime} \exp \left(-i \mathbf{k}^{\prime} \cdot \mathbf{r}^{\prime}\right) V\left(\mathbf{r}^{\prime}\right) \psi\left(\mathbf{k}, \mathbf{r}^{\prime}\right)$$

  [You may assume that

  $$\mathcal{G}(k ; \mathbf{r})=-\frac{1}{4 \pi|\mathbf{r}|} \exp (i k|\mathbf{r}|)$$

  is the Green's function for $\nabla^{2}+k^{2}$ which obeys the appropriate boundary conditions for a scattering solution.]

  Now suppose $V(\mathbf{r})=\lambda U(\mathbf{r})$, where $\lambda \ll 1$ is a dimensionless constant. Determine the first two non-zero terms in the expansion of $f\left(\mathbf{k}, \mathbf{k}^{\prime}\right)$ in powers of $\lambda$, giving each term explicitly as an integral over one or more position variables $\mathbf{r}, \mathbf{r}^{\prime}, \ldots$

  Evaluate the contribution to $f\left(\mathbf{k}, \mathbf{k}^{\prime}\right)$ of order $\lambda$ in the case $U(\mathbf{r})=\delta(|\mathbf{r}|-a)$, expressing the answer as a function of $a, k$ and the scattering angle $\theta$ (defined so that $\left.\mathbf{k} \cdot \mathbf{k}^{\prime}=k^{2} \cos \theta\right)$.

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• Paper 3, Section II, J

  Applied Probability

  Part II, 2016

  (a) State the thinning and superposition properties of a Poisson process on $\mathbb{R}_{+}$. Prove the superposition property.

  (b) A bi-infinite Poisson process $\left(N_{t}: t \in \mathbb{R}\right)$ with $N_{0}=0$ is a process with independent and stationary increments over $\mathbb{R}$. Moreover, for all $-\infty<s \leqslant t<\infty$, the increment $N_{t}-N_{s}$ has the Poisson distribution with parameter $\lambda(t-s)$. Prove that such a process exists.

  (c) Let $N$ be a bi-infinite Poisson process on $\mathbb{R}$ of intensity $\lambda$. We identify it with the set of points $\left(S_{n}\right)$ of discontinuity of $N$, i.e., $N[s, t]=\sum_{n} \mathbf{l}\left(S_{n} \in[s, t]\right)$. Show that if we shift all the points of $N$ by the same constant $c$, then the resulting process is also a Poisson process of intensity $\lambda$.

  Now suppose we shift every point of $N$ by $+1$ or $-1$ with equal probability. Show that the final collection of points is still a Poisson process of intensity $\lambda$. [You may assume the thinning and superposition properties for the bi-infinite Poisson process.]

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• Paper 3, Section II, A

  Applications of Quantum Mechanics

  Part II, 2016

  (a) A spinless charged particle moves in an electromagnetic field defined by vector and scalar potentials $\mathbf{A}(\mathbf{x}, t)$ and $\phi(\mathbf{x}, t)$. The wavefunction $\psi(\mathbf{x}, t)$ for the particle satisfies the time-dependent Schrödinger equation with Hamiltonian

  $$\hat{H}_{0}=\frac{1}{2 m}(-i \hbar \boldsymbol{\nabla}+e \mathbf{A}) \cdot(-i \hbar \boldsymbol{\nabla}+e \mathbf{A})-e \phi .$$

  Consider a gauge transformation

  $$\mathbf{A} \rightarrow \tilde{\mathbf{A}}=\mathbf{A}+\nabla f, \quad \phi \rightarrow \tilde{\phi}=\phi-\frac{\partial f}{\partial t}, \quad \psi \rightarrow \tilde{\psi}=\exp \left(-\frac{i e f}{\hbar}\right) \psi$$

  for some function $f(\mathbf{x}, t)$. Define covariant derivatives with respect to space and time, and show that $\tilde{\psi}$ satisfies the Schrödinger equation with potentials $\tilde{\mathbf{A}}$ and $\tilde{\phi}$.

  (b) Suppose that in part (a) the magnetic field has the form $\mathbf{B}=\boldsymbol{\nabla} \times \mathbf{A}=(0,0, B)$, where $B$ is a constant, and that $\phi=0$. Find a suitable $\mathbf{A}$ with $A_{y}=A_{z}=0$ and determine the energy levels of the Hamiltonian $\hat{H}_{0}$ when the $z$-component of the momentum of the particle is zero. Suppose in addition that the particle is constrained to lie in a rectangular region of area $\mathcal{A}$ in the $(x, y)$-plane. By imposing periodic boundary conditions in the $x$-direction, estimate the degeneracy of each energy level. [You may use without proof results for a quantum harmonic oscillator, provided they are clearly stated.]

  (c) An electron is a charged particle of spin $\frac{1}{2}$ with a two-component wavefunction $\psi(\mathbf{x}, t)$ governed by the Hamiltonian

  $$\hat{H}=\hat{H}_{0} \mathbb{I}_{2}+\frac{e \hbar}{2 m} \mathbf{B} \cdot \boldsymbol{\sigma}$$

  where $\mathbb{I}_{2}$ is the $2 \times 2$ unit matrix and $\sigma=\left(\sigma_{1}, \sigma_{2}, \sigma_{3}\right)$ denotes the Pauli matrices. Find the energy levels for an electron in the constant magnetic field defined in part (b), assuming as before that the $z$-component of the momentum of the particle is zero.

  Consider $N$ such electrons confined to the rectangular region defined in part (b). Ignoring interactions between the electrons, show that the ground state energy of this system vanishes for $N$ less than some integer $N_{\max }$ which you should determine. Find the ground state energy for $N=(2 p+1) N_{\max }$, where $p$ is a positive integer.

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• Paper 2, Section II, J

  Applied Probability

  Part II, 2016

  (a) Define an $M / M / \infty$ queue and write without proof its stationary distribution. State Burke's theorem for an $M / M / \infty$ queue.

  (b) Let $X$ be an $M / M / \infty$ queue with arrival rate $\lambda$ and service rate $\mu$ started from the stationary distribution. For each $t$, denote by $A_{1}(t)$ the last time before $t$ that a customer departed the queue and $A_{2}(t)$ the first time after $t$ that a customer departed the queue. If there is no arrival before time $t$, then we set $A_{1}(t)=0$. What is the limit as $t \rightarrow \infty$ of $\mathbb{E}\left[A_{2}(t)-A_{1}(t)\right]$ ? Explain.

  (c) Consider a system of $N$ queues serving a finite number $K$ of customers in the following way: at station $1 \leqslant i \leqslant N$, customers are served immediately and the service times are independent exponentially distributed with parameter $\mu_{i}$; after service, each customer goes to station $j$ with probability $p_{i j}>0$. We assume here that the system is closed, i.e., $\sum_{j} p_{i j}=1$ for all $1 \leqslant i \leqslant N$.

  Let $S=\left\{\left(n_{1}, \ldots, n_{N}\right): n_{i} \in \mathbb{N}, \sum_{i=1}^{N} n_{i}=K\right\}$ be the state space of the Markov chain. Write down its $Q$-matrix. Also write down the $Q$-matrix $R$ corresponding to the position in the network of one customer (that is, when $K=1$ ). Show that there is a unique distribution $\left(\lambda_{i}\right)_{1 \leqslant i \leqslant N}$ such that $\lambda R=0$. Show that

  $$\pi(n)=C_{N} \prod_{i=1}^{N} \frac{\lambda_{i}^{n_{i}}}{n_{i} !}, \quad n=\left(n_{1}, \ldots, n_{N}\right) \in S$$

  defines an invariant measure for the chain. Are the queue lengths independent at equilibrium?

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• Paper 4, Section II, J

  Applied Probability

  Part II, 2016

  (a) Give the definition of a renewal process. Let $\left(N_{t}\right)_{t \geqslant 0}$ be a renewal process associated with $\left(\xi_{i}\right)$ with $\mathbb{E} \xi_{1}=1 / \lambda<\infty$. Show that almost surely

  $$\frac{N_{t}}{t} \rightarrow \lambda \quad \text { as } t \rightarrow \infty$$

  (b) Give the definition of Kingman's $n$-coalescent. Let $\tau$ be the first time that all blocks have coalesced. Find an expression for $\mathbb{E} e^{-q \tau}$. Let $L_{n}$ be the total length of the branches of the tree, i.e., if $\tau_{i}$ is the first time there are $i$ lineages, then $L_{n}=$ $\sum_{i=2}^{n} i\left(\tau_{i-1}-\tau_{i}\right)$. Show that $\mathbb{E} L_{n} \sim 2 \log n$ as $n \rightarrow \infty$. Show also that $\operatorname{Var}\left(L_{n}\right) \leqslant C$ for all $n$, where $C$ is a positive constant, and that in probability

  $$\frac{L_{n}}{\mathbb{E} L_{n}} \rightarrow 1 \quad \text { as } n \rightarrow \infty$$

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• Paper 1, Section II, J

  Applied Probability

  Part II, 2016

  (a) Define a continuous-time Markov chain $X$ with infinitesimal generator $Q$ and jump chain $Y$.

  (b) Prove that if a state $x$ is transient for $Y$, then it is transient for $X$.

  (c) Prove or provide a counterexample to the following: if $x$ is positive recurrent for $X$, then it is positive recurrent for $Y$.

  (d) Consider the continuous-time Markov chain $\left(X_{t}\right)_{t \geqslant 0}$ on $\mathbb{Z}$ with non-zero transition rates given by

  $$q(i, i+1)=2 \cdot 3^{|i|}, \quad q(i, i)=-3^{|i|+1} \quad \text { and } \quad q(i, i-1)=3^{|i|}$$

  Determine whether $X$ is transient or recurrent. Let $T_{0}=\inf \left\{t \geqslant J_{1}: X_{t}=0\right\}$, where $J_{1}$ is the first jump time. Does $X$ have an invariant distribution? Justify your answer. Calculate $\mathbb{E}_{0}\left[T_{0}\right]$.

  (e) Let $X$ be a continuous-time random walk on $\mathbb{Z}^{d}$ with $q(x)=\|x\|^{\alpha} \wedge 1$ and $q(x, y)=q(x) /(2 d)$ for all $y \in \mathbb{Z}^{d}$ with $\|y-x\|=1$. Determine for which values of $\alpha$ the walk is transient and for which it is recurrent. In the recurrent case, determine the range of $\alpha$ for which it is also positive recurrent. [Here $\|x\|$ denotes the Euclidean norm of $x$.]

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• Paper 3, Section II, C

  Asymptotic Methods

  Part II, 2016

  Consider the integral

  $$I(x)=\int_{0}^{1} \frac{1}{\sqrt{t(1-t)}} \exp [i x f(t)] d t$$

  for real $x>0$, where $f(t)=t^{2}+t$. Find and sketch, in the complex $t$-plane, the paths of steepest descent through the endpoints $t=0$ and $t=1$ and through any saddle point(s). Obtain the leading order term in the asymptotic expansion of $I(x)$ for large positive $x$. What is the order of the next term in the expansion? Justify your answer.

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• Paper 2, Section II, C

  Asymptotic Methods

  Part II, 2016

  What is meant by the asymptotic relation

  $$f(z) \sim g(z) \quad \text { as } \quad z \rightarrow z_{0}, \operatorname{Arg}\left(z-z_{0}\right) \in\left(\theta_{0}, \theta_{1}\right) ?$$

  Show that

  $$\sinh \left(z^{-1}\right) \sim \frac{1}{2} \exp \left(z^{-1}\right) \quad \text { as } \quad z \rightarrow 0, \operatorname{Arg} z \in(-\pi / 2, \pi / 2),$$

  and find the corresponding result in the sector $\operatorname{Arg} z \in(\pi / 2,3 \pi / 2)$.

  What is meant by the asymptotic expansion

  $$f(z) \sim \sum_{j=0}^{\infty} c_{j}\left(z-z_{0}\right)^{j} \quad \text { as } \quad z \rightarrow z_{0}, \operatorname{Arg}\left(z-z_{0}\right) \in\left(\theta_{0}, \theta_{1}\right) ?$$

  Show that the coefficients $\left\{c_{j}\right\}_{j=0}^{\infty}$ are determined uniquely by $f$. Show that if $f$ is analytic at $z_{0}$, then its Taylor series is an asymptotic expansion for $f$ as $z \rightarrow z_{0}\left(\right.$ for any $\left.\operatorname{Arg}\left(z-z_{0}\right)\right)$.

  Show that

  $$u(x, t)=\int_{-\infty}^{\infty} \exp \left(-i k^{2} t+i k x\right) f(k) d k$$

  defines a solution of the equation $i \partial_{t} u+\partial_{x}^{2} u=0$ for any smooth and rapidly decreasing function $f$. Use the method of stationary phase to calculate the leading-order behaviour of $u(\lambda t, t)$ as $t \rightarrow+\infty$, for fixed $\lambda$.

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• Paper 4, Section II, C

  Asymptotic Methods

  Part II, 2016

  Consider the equation

  $$\epsilon^{2} \frac{d^{2} y}{d x^{2}}=Q(x) y$$

  where $\epsilon>0$ is a small parameter and $Q(x)$ is smooth. Search for solutions of the form

  $$y(x)=\exp \left[\frac{1}{\epsilon}\left(S_{0}(x)+\epsilon S_{1}(x)+\epsilon^{2} S_{2}(x)+\cdots\right)\right],$$

  and, by equating powers of $\epsilon$, obtain a collection of equations for the $\left\{S_{j}(x)\right\}_{j=0}^{\infty}$ which is formally equivalent to (1). By solving explicitly for $S_{0}$ and $S_{1}$ derive the Liouville- Green approximate solutions $y^{L G}(x)$ to (1).

  For the case $Q(x)=-V(x)$, where $V(x) \geqslant V_{0}$ and $V_{0}$ is a positive constant, consider the eigenvalue problem

  $$\frac{d^{2} y}{d x^{2}}+E V(x) y=0, \quad y(0)=y(\pi)=0$$

  Show that any eigenvalue $E$ is necessarily positive. Solve the eigenvalue problem exactly when $V(x)=V_{0}$.

  Obtain Liouville-Green approximate eigenfunctions $y_{n}^{L G}(x)$ for (2) with $E \gg 1$, and give the corresponding Liouville Green approximation to the eigenvalues $E_{n}^{L G}$. Compare your results to the exact eigenvalues and eigenfunctions in the case $V(x)=V_{0}$, and comment on this.

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• Paper 4, Section I, $4 \mathrm{~F}$

  Automata and Formal Languages

  Part II, 2016

  (a) Construct a register machine to compute the function $f(m, n):=m+n$. State the relationship between partial recursive functions and partial computable functions. Show that the function $g(m, n):=m n$ is partial recursive.

  (b) State Rice's theorem. Show that the set $A:=\left\{n \in \mathbb{N}|| W_{n} \mid>7\right\}$ is recursively enumerable but not recursive.

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• Paper 3, Section I, $4 F$

  Automata and Formal Languages

  Part II, 2016

  (a) Define what it means for a context-free grammar (CFG) to be in Chomsky normal form (CNF). Can a CFG in CNF ever define a language containing $\epsilon$ ? If $G_{\text {Chom }}$ denotes the result of converting an arbitrary CFG $G$ into one in CNF, state the relationship between $\mathcal{L}(G)$ and $\mathcal{L}\left(G_{\text {Chom }}\right)$.

  (b) Let $G$ be a CFG in CNF, and let $w \in \mathcal{L}(G)$ be a word of length $|w|=n>0$. Show that every derivation of $w$ in $G$ requires precisely $2 n-1$ steps. Using this, give an algorithm that, on input of any word $v$ on the terminals of $G$, decides if $v \in \mathcal{L}(G)$ or not.

  (c) Convert the following CFG $G$ into a grammar in CNF:

  $$S \rightarrow a S b|S S| \epsilon$$

  Does $\mathcal{L}(G)=\mathcal{L}\left(G_{\text {Chom }}\right)$ in this case? Justify your answer.

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• Paper 2, Section $\mathbf{I}$, $4 F$

  Automata and Formal Languages

  Part II, 2016

  (a) Which of the following are regular languages? Justify your answers.

  (i) $\left\{w \in\{a, b\}^{*} \mid w\right.$ is a nonempty string of alternating $a$ 's and $b$ 's $\}$.

  (ii) $\left\{w a b w \mid w \in\{a, b\}^{*}\right\}$.

  (b) Write down a nondeterministic finite-state automaton with $\epsilon$-transitions which accepts the language given by the regular expression $(\mathbf{a}+\mathbf{b})^{*}(\mathbf{b} \mathbf{b}+\mathbf{a}) \mathbf{b}$. Describe in words what this language is.

  (c) Is the following language regular? Justify your answer.

  $$\left\{w \in\{a, b\}^{*} \mid w \text { does not end in } a b \text { or } b b b\right\}$$

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• Paper 1, Section I, $4 \mathrm{~F}$

  Automata and Formal Languages

  Part II, 2016

  State the pumping lemma for context-free languages (CFLs). Which of the following are CFLs? Justify your answers.

  (i) $\left\{a^{2 n} b^{3 n} \mid n \geqslant 3\right\}$.

  (ii) $\left\{a^{2 n} b^{3 n} c^{5 n} \mid n \geqslant 0\right\}$.

  (iii) $\left\{a^{p} \mid p\right.$ is a prime $\}$.

  Let $L, M$ be CFLs. Show that $L \cup M$ is also a CFL.

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• Paper 3, Section II, F

  Automata and Formal Languages

  Part II, 2016

  (a) Let $D=\left(Q, \Sigma, \delta, q_{0}, F\right)$ be a deterministic finite-state automaton. Define the extended transition function $\hat{\delta}: Q \times \Sigma^{*} \rightarrow Q$, and the language accepted by $D$, denoted $\mathcal{L}(D)$. Let $u, v \in \Sigma^{*}$, and $p \in Q$. Prove that $\hat{\delta}(p, u v)=\hat{\delta}(\hat{\delta}(p, u), v)$.

  (b) Let $\sigma_{1}, \sigma_{2}, \ldots, \sigma_{k} \in \Sigma$ where $k \geqslant|Q|$, and let $p \in Q$.

  (i) Show that there exist $0 \leqslant i<j \leqslant k$ such that $\hat{\delta}\left(p, \sigma_{1} \cdots \sigma_{i}\right)=\hat{\delta}\left(p, \sigma_{1} \cdots \sigma_{j}\right)$, where we interpret $\sigma_{1} \cdots \sigma_{i}$ as $\epsilon$ if $i=0$.

  (ii) Show that $\hat{\delta}\left(p, \sigma_{1} \cdots \sigma_{i} \sigma_{j+1} \cdots \sigma_{k}\right)=\hat{\delta}\left(p, \sigma_{1} \cdots \sigma_{k}\right)$.

  (iii) Show that $\hat{\delta}\left(p, \sigma_{1} \cdots \sigma_{i}\left(\sigma_{i+1} \cdots \sigma_{j}\right)^{t} \sigma_{j+1} \cdots \sigma_{k}\right)=\hat{\delta}\left(p, \sigma_{1} \cdots \sigma_{k}\right)$ for all $t \geqslant 1$.

  (c) Prove the following version of the pumping lemma. Suppose $w \in \mathcal{L}(D)$, with $|w| \geqslant|Q|$. Then $w$ can be broken up into three words $w=x y z$ such that $y \neq \epsilon,|x y| \leqslant|Q|$, and for all $t \geqslant 0$, the word $x y^{t} z$ is also in $\mathcal{L}(D)$.

  (d) Hence show that

  (i) if $\mathcal{L}(D)$ contains a word of length at least $|Q|$, then it contains infinitely many words;

  (ii) if $\mathcal{L}(D) \neq \emptyset$, then it contains a word of length less than $|Q|$;

  (iii) if $\mathcal{L}(D)$ contains all words in $\Sigma^{*}$ of length less than $|Q|$, then $\mathcal{L}(D)=\Sigma^{*}$.

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• Paper 1, Section II, F

  Automata and Formal Languages

  Part II, 2016

  (a) Define a recursive set and a recursively enumerable (r.e.) set. Prove that $E \subseteq \mathbb{N}$ is recursive if and only if both $E$ and $\mathbb{N} \backslash E$ are r.e.

  (b) Define the halting set $\mathbb{K}$. Prove that $\mathbb{K}$ is r.e. but not recursive.

  (c) Let $E_{1}, E_{2}, \ldots, E_{n}$ be r.e. sets. Prove that $\bigcup_{i=1}^{n} E_{i}$ and $\bigcap_{i=1}^{n} E_{i}$ are r.e. Show by an example that the union of infinitely many r.e. sets need not be r.e.

  (d) Let $E$ be a recursive set and $f: \mathbb{N} \rightarrow \mathbb{N}$ a (total) recursive function. Prove that the set $\{f(n) \mid n \in E\}$ is r.e. Is it necessarily recursive? Justify your answer.

  [Any use of Church's thesis in your answer should be explicitly stated.]

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• Paper 4, Section I, E

  Classical Dynamics

  Part II, 2016

  Using conservation of angular momentum $\mathbf{L}=L_{a} \mathbf{e}_{a}$ in the body frame, derive the Euler equations for a rigid body:

  $$I_{1} \dot{\omega}_{1}+\left(I_{3}-I_{2}\right) \omega_{2} \omega_{3}=0, \quad I_{2} \dot{\omega}_{2}+\left(I_{1}-I_{3}\right) \omega_{3} \omega_{1}=0, \quad I_{3} \dot{\omega}_{3}+\left(I_{2}-I_{1}\right) \omega_{1} \omega_{2}=0$$

  [You may use the formula $\dot{\mathbf{e}}_{a}=\boldsymbol{\omega} \wedge \mathbf{e}_{a}$ without proof.]

  Assume that the principal moments of inertia satisfy $I_{1}<I_{2}<I_{3}$. Determine whether a rotation about the principal 3-axis leads to stable or unstable perturbations.

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• Paper 1, Section I, E

  Classical Dynamics

  Part II, 2016

  Consider a one-parameter family of transformations $q_{i}(t) \mapsto Q_{i}(s, t)$ such that $Q_{i}(0, t)=q_{i}(t)$ for all time $t$, and

  $$\frac{\partial}{\partial s} L\left(Q_{i}, \dot{Q}_{i}, t\right)=0$$

  where $L$ is a Lagrangian and a dot denotes differentiation with respect to $t$. State and prove Noether's theorem.

  Consider the Lagrangian

  $$L=\frac{1}{2}\left(\dot{x}^{2}+\dot{y}^{2}+\dot{z}^{2}\right)-V(x+y, y+z),$$

  where the potential $V$ is a function of two variables. Find a continuous symmetry of this Lagrangian and construct the corresponding conserved quantity. Use the Euler-Lagrange equations to explicitly verify that the function you have constructed is independent of $t$.

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• Paper 2, Section I, E

  Classical Dynamics

  Part II, 2016

  Consider the Lagrangian

  $$L=A\left(\dot{\theta}^{2}+\dot{\phi}^{2} \sin ^{2} \theta\right)+B(\dot{\psi}+\dot{\phi} \cos \theta)^{2}-C(\cos \theta)^{k}$$

  where $A, B, C$ are positive constants and $k$ is a positive integer. Find three conserved quantities and show that $u=\cos \theta$ satisfies

  $$\dot{u}^{2}=f(u)$$

  where $f(u)$ is a polynomial of degree $k+2$ which should be determined.

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• Paper 3, Section I, E

  Classical Dynamics

  Part II, 2016

  Consider a six-dimensional phase space with coordinates $\left(q_{i}, p_{i}\right)$ for $i=1,2,3$. Compute the Poisson brackets $\left\{L_{i}, L_{j}\right\}$, where $L_{i}=\epsilon_{i j k} q_{j} p_{k}$.

  Consider the Hamiltonian

  $$H=\frac{1}{2}|\mathbf{p}|^{2}+V(|\mathbf{q}|)$$

  and show that the resulting Hamiltonian system admits three Poisson-commuting independent first integrals.

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• Paper 2, Section II, E

  Classical Dynamics

  Part II, 2016

  Define what it means for the transformation $\mathbb{R}^{2 n} \rightarrow \mathbb{R}^{2 n}$ given by

  $$\left(q_{i}, p_{i}\right) \mapsto\left(Q_{i}\left(q_{j}, p_{j}\right), P_{i}\left(q_{j}, p_{j}\right)\right), \quad i, j=1, \ldots, n$$

  to be canonical. Show that a transformation is canonical if and only if

  $$\left\{Q_{i}, Q_{j}\right\}=0, \quad\left\{P_{i}, P_{j}\right\}=0, \quad\left\{Q_{i}, P_{j}\right\}=\delta_{i j}$$

  Show that the transformation $\mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ given by

  $$Q=q \cos \epsilon-p \sin \epsilon, \quad P=q \sin \epsilon+p \cos \epsilon$$

  is canonical for any real constant $\epsilon$. Find the corresponding generating function.

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• Paper 4, Section II, $14 \mathrm{E}$

  Classical Dynamics

  Part II, 2016

  A particle of unit mass is attached to one end of a light, stiff rod of length $\ell$. The other end of the rod is held at a fixed position, such that the rod is free to swing in any direction. Write down the Lagrangian for the system giving a clear definition of any angular variables you introduce. [You should assume the acceleration $g$ is constant.]

  Find two independent constants of the motion.

  The particle is projected horizontally with speed $v$ from a point where the rod lies at an angle $\alpha$ to the downward vertical, with $0<\alpha<\pi / 2$. In terms of $\ell, g$ and $\alpha$, find the critical speed $v_{c}$ such that the particle always remains at its initial height.

  The particle is now projected horizontally with speed $v_{c}$ but from a point at angle $\alpha+\delta \alpha$ to the vertical, where $\delta \alpha / \alpha \ll 1$. Show that the height of the particle oscillates, and find the period of oscillation in terms of $\ell, g$ and $\alpha$.

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• Paper 1, Section I, G

  Coding and Cryptography

  Part II, 2016

  Find the average length of an optimum decipherable binary code for a source that emits five words with probabilities

  $$0.25,0.15,0.15,0.2,0.25$$

  Show that, if a source emits $N$ words (with $N \geqslant 2$ ), and if $l_{1}, \ldots, l_{N}$ are the lengths of the codewords in an optimum encoding over the binary alphabet, then

  $$l_{1}+\cdots+l_{N} \leqslant \frac{1}{2}\left(N^{2}+N-2\right) .$$

  [You may assume that an optimum encoding can be given by a Huffman encoding.]

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• Paper 2, Section I, G

  Coding and Cryptography

  Part II, 2016

  Show that the binary channel with channel matrix

  $$\left(\begin{array}{ll} 1 & 0 \\ \frac{1}{2} & \frac{1}{2} \end{array}\right)$$

  has capacity $\log 5-2$.

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• Paper 3, Section I, $3 G$

  Coding and Cryptography

  Part II, 2016

  Describe in words the unicity distance of a cryptosystem.

  Denote the cryptosystem by $\langle M, K, C\rangle$, in the usual way, and let $m \in M$ and $k \in K$ be random variables and $c=e(m, k)$. The unicity distance $U$ is formally defined to be the least $n>0$ such that $H\left(k \mid c^{(n)}\right)=0$. Derive the formula

  $$U=\frac{\log |K|}{\log |\Sigma|-H}$$

  where $H=H(m)$, and $\Sigma$ is the alphabet of the ciphertext. Make clear any assumptions you make.

  The redundancy of a language is given by $R=1-\frac{H}{\log |\Sigma|}$. If a language has zero redundancy what is the unicity of any cryptosystem?

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• Paper 4, Section I, G

  Coding and Cryptography

  Part II, 2016

  Describe the Rabin-Williams scheme for coding a message $x$ as $x^{2}$ modulo a certain $N$. Show that, if $N$ is chosen appropriately, breaking this code is equivalent to factorising the product of two primes.

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• Paper 1, Section II, G

  Coding and Cryptography

  Part II, 2016

  What does it mean to say a binary code $C$ has length $n$, size $m$ and minimum distance d?

  Let $A(n, d)$ be the largest value of $m$ for which there exists an $[n, m, d]$-code. Prove that

  $$\frac{2^{n}}{V(n, d-1)} \leqslant A(n, d) \leqslant \frac{2^{n}}{V(n,\lfloor(d-1) / 2\rfloor)}$$

  where

  $$V(n, r)=\sum_{j=0}^{r}\left(\begin{array}{l} n \\ j \end{array}\right)$$

  Another bound for $A(n, d)$ is the Singleton bound given by

  $$A(n, d) \leqslant 2^{n-d+1}$$

  Prove the Singleton bound and give an example of a linear code of length $n>1$ that satisfies $A(n, d)=2^{n-d+1}$.

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• Paper 2, Section II, G

  Coding and Cryptography

  Part II, 2016

  Define a $B C H$ code of length $n$, where $n$ is odd, over the field of 2 elements with design distance $\delta$. Show that the minimum weight of such a code is at least $\delta$. [Results about the Vandermonde determinant may be quoted without proof, provided they are stated clearly.]

  Let $\omega \in \mathbb{F}_{16}$ be a root of $X^{4}+X+1$. Let $C$ be the $\mathrm{BCH}$ code of length 15 with defining set $\left\{\omega, \omega^{2}, \omega^{3}, \omega^{4}\right\}$. Find the generator polynomial of $C$ and the rank of $C$. Determine the error positions of the following received words:

  (i) $r(X)=1+X^{6}+X^{7}+X^{8}$,

  (ii) $r(X)=1+X+X^{4}+X^{5}+X^{6}+X^{9}$.

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• Paper 1, Section I, C

  Cosmology

  Part II, 2016

  The expansion scale factor, $a(t)$, for an isotropic and spatially homogeneous universe containing material with pressure $p$ and mass density $\rho$ obeys the equations

  $$\begin{gathered} \dot{\rho}+3(\rho+p) \frac{\dot{a}}{a}=0, \\ \left(\frac{\dot{a}}{a}\right)^{2}=\frac{8 \pi G \rho}{3}-\frac{k}{a^{2}}+\frac{\Lambda}{3}, \end{gathered}$$

  where the speed of light is set equal to unity, $G$ is Newton's constant, $k$ is a constant equal to 0 or $\pm 1$, and $\Lambda$ is the cosmological constant. Explain briefly the interpretation of these equations.

  Show that these equations imply

  $$\frac{\ddot{a}}{a}=-\frac{4 \pi G(\rho+3 p)}{3}+\frac{\Lambda}{3} .$$

  Hence show that a static solution with constant $a=a_{\mathrm{s}}$ exists when $p=0$ if

  $$\Lambda=4 \pi G \rho=\frac{k}{a_{\mathrm{s}}^{2}}$$

  What must the value of $k$ be, if the density $\rho$ is non-zero?

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• Paper 2, Section I, C

  Cosmology

  Part II, 2016

  A spherical cloud of mass $M$ has radius $r(t)$ and initial radius $r(0)=R$. It contains material with uniform mass density $\rho(t)$, and zero pressure. Ignoring the cosmological constant, show that if it is initially at rest at $t=0$ and the subsequent gravitational collapse is governed by Newton's law $\ddot{r}=-G M / r^{2}$, then

  $$\dot{r}^{2}=2 G M\left(\frac{1}{r}-\frac{1}{R}\right) .$$

  Suppose $r$ is given parametrically by

  $$r=R \cos ^{2} \theta$$

  where $\theta=0$ at $t=0$. Derive a relation between $\theta$ and $t$ and hence show that the cloud collapses to radius $r=0$ at

  $$t=\sqrt{\frac{3 \pi}{32 G \rho_{0}}},$$

  where $\rho_{0}$ is the initial mass density of the cloud.

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• Paper 3, Section I, C

  Cosmology

  Part II, 2016

  A universe contains baryonic matter with background density $\rho_{B}(t)$ and density inhomogeneity $\delta_{B}(\mathbf{x}, t)$, together with non-baryonic dark matter with background density $\rho_{D}(t)$ and density inhomogeneity $\delta_{D}(\mathbf{x}, t)$. After the epoch of radiation-matter density equality, $t_{\mathrm{eq}}$, the background dynamics are governed by

  $$H=\frac{2}{3 t} \quad \text { and } \quad \rho_{D}=\frac{1}{6 \pi G t^{2}}$$

  where $H$ is the Hubble parameter.

  The dark-matter density is much greater than the baryonic density $\left(\rho_{D} \gg \rho_{B}\right)$ and so the time-evolution of the coupled density perturbations, at any point $\mathbf{x}$, is described by the equations

  $$\begin{aligned} &\ddot{\delta}_{B}+2 H \dot{\delta}_{B}=4 \pi G \rho_{D} \delta_{D} \\ &\ddot{\delta}_{D}+2 H \dot{\delta}_{D}=4 \pi G \rho_{D} \delta_{D} \end{aligned}$$

  Show that

  $$\delta_{D}=\frac{\alpha}{t}+\beta t^{2 / 3}$$

  where $\alpha$ and $\beta$ are independent of time. Neglecting modes in $\delta_{D}$ and $\delta_{B}$ that decay with increasing time, show that the baryonic density inhomogeneity approaches

  $$\delta_{B}=\beta t^{2 / 3}+\gamma$$

  where $\gamma$ is independent of time.

  Briefly comment on the significance of your calculation for the growth of baryonic density inhomogeneities in the early universe.

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• Paper 4, Section I, C

  Cosmology

  Part II, 2016

  The external gravitational potential $\Phi(r)$ due to a thin spherical shell of radius $a$ and mass per unit area $\sigma$, centred at $r=0$, will equal the gravitational potential due to a point mass $M$ at $r=0$, at any distance $r>a$, provided

  $$\frac{M r \Phi(r)}{2 \pi \sigma a}+K(a) r=\int_{r-a}^{r+a} R \Phi(R) d R$$

  where $K(a)$ depends on the radius of the shell. For which values of $q$ does this equation have solutions of the form $\Phi(r)=C r^{q}$, where $C$ is constant? Evaluate $K(a)$ in each case and find the relation between the mass of the shell and $M$.

  Hence show that the general gravitational force

  $$F(r)=\frac{A}{r^{2}}+B r$$

  has a potential satisfying $(*)$. What is the cosmological significance of the constant $B$ ?

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• Paper 3, Section II, C

  Cosmology

  Part II, 2016

  The early universe is described by equations (with units such that $c=8 \pi G=\hbar=1$ )

  $$3 H^{2}=\rho, \quad \dot{\rho}+3 H(\rho+p)=0$$

  where $H=\dot{a} / a$. The universe contains only a self-interacting scalar field $\phi$ with interaction potential $V(\phi)$ so that the density and pressure are given by

  $$\begin{aligned} \rho &=\frac{1}{2} \dot{\phi}^{2}+V(\phi) \\ p &=\frac{1}{2} \dot{\phi}^{2}-V(\phi) . \end{aligned}$$

  Show that

  $$\ddot{\phi}+3 H \dot{\phi}+V^{\prime}(\phi)=0$$

  Explain the slow-roll approximation and apply it to equations (1) and (2) to show that it leads to

  $$\sqrt{3} \int \frac{\sqrt{V}}{V^{\prime}} d \phi=-t+\text { const. }$$

  If $V(\phi)=\frac{1}{4} \lambda \phi^{4}$ with $\lambda$ a positive constant and $\phi(0)=\phi_{0}$, show that

  $$\phi(t)=\phi_{0} \exp \left[-\sqrt{\frac{4 \lambda}{3}} t\right]$$

  and that, for small $t$, the scale factor $a(t)$ expands to leading order in $t$ as

  $$a(t) \propto \exp \left[\sqrt{\frac{\lambda}{12}} \phi_{0}^{2} t\right]$$

  Comment on the relevance of this result for inflationary cosmology.

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• Paper 1, Section II, C

  Cosmology

  Part II, 2016

  The distribution function $f(\mathbf{x}, \mathbf{p}, t)$ gives the number of particles in the universe with position in $(\mathbf{x}, \mathbf{x}+\delta \mathbf{x})$ and momentum in $(\mathbf{p}, \mathbf{p}+\delta \mathbf{p})$ at time $t$. It satisfies the boundary condition that $f \rightarrow 0$ as $|\mathbf{x}| \rightarrow \infty$ and as $|\mathbf{p}| \rightarrow \infty$. Its evolution obeys the Boltzmann equation

  $$\frac{\partial f}{\partial t}+\frac{\partial f}{\partial \mathbf{p}} \cdot \frac{d \mathbf{p}}{d t}+\frac{\partial f}{\partial \mathbf{x}} \cdot \frac{d \mathbf{x}}{d t}=\left[\frac{d f}{d t}\right]_{\mathrm{col}}$$

  where the collision term $\left[\frac{d f}{d t}\right]_{c o l}$ describes any particle production and annihilation that occurs.

  The universe expands isotropically and homogeneously with expansion scale factor $a(t)$, so the momenta evolve isotropically with magnitude $p \propto a^{-1}$. Show that the Boltzmann equation simplifies to

  $$\frac{\partial f}{\partial t}-\frac{\dot{a}}{a} \mathbf{p} \cdot \frac{\partial f}{\partial \mathbf{p}}=\left[\frac{d f}{d t}\right]_{\mathrm{col}}$$

  The number densities $n$ of particles and $\bar{n}$ of antiparticles are defined in terms of their distribution functions $f$ and $\bar{f}$, and momenta $p$ and $\bar{p}$, by

  $$n=\int_{0}^{\infty} f 4 \pi p^{2} d p \quad \text { and } \quad \bar{n}=\int_{0}^{\infty} \bar{f} 4 \pi \bar{p}^{2} d \bar{p}$$

  and the collision term may be assumed to be of the form

  $$\left[\frac{d f}{d t}\right]_{\mathrm{col}}=-\langle\sigma v\rangle \int_{0}^{\infty} \bar{f} f 4 \pi \bar{p}^{2} d \bar{p}+R$$

  where $\langle\sigma v\rangle$ determines the annihilation cross-section of particles by antiparticles and $R$ is the production rate of particles.

  By integrating equation $(*)$ with respect to the momentum $\mathbf{p}$ and assuming that $\langle\sigma v\rangle$ is a constant, show that

  $$\frac{d n}{d t}+3 \frac{\dot{a}}{a} n=-\langle\sigma v\rangle n \bar{n}+Q$$

  where $Q=\int_{0}^{\infty} R 4 \pi p^{2} d p$. Assuming the same production rate $R$ for antiparticles, write down the corresponding equation satisfied by $\bar{n}$ and show that

  $$(n-\bar{n}) a^{3}=\mathrm{constant}$$

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• Paper 3, Section II, G

  Differential Geometry

  Part II, 2016

  Explain what it means for an embedded surface $S$ in $\mathbf{R}^{3}$ to be minimal. What is meant by an isothermal parametrization $\phi: U \rightarrow V \subset \mathbf{R}^{3}$ of an embedded surface $V \subset \mathbf{R}^{3}$ ? Prove that if $\phi$ is isothermal then $\phi(U)$ is minimal if and only if the components of $\phi$ are harmonic functions on $U$. [You may assume the formula for the mean curvature of a parametrized embedded surface,

  $$H=\frac{e G-2 f F+g E}{2\left(E G-F^{2}\right)}$$

  where $E, F, G$ (respectively $e, f, g$ ) are the coefficients of the first (respectively second) fundamental forms.]

  Let $S$ be an embedded connected minimal surface in $\mathbf{R}^{3}$ which is closed as a subset of $\mathbf{R}^{3}$, and let $\Pi \subset \mathbf{R}^{3}$ be a plane which is disjoint from $S$. Assuming that local isothermal parametrizations always exist, show that if the Euclidean distance between $S$ and $\Pi$ is attained at some point $P \in S$, i.e. $d(P, \Pi)=\inf _{Q \in S} d(Q, \Pi)$, then $S$ is a plane parallel to 川.

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• Paper 4, Section II, G

  Differential Geometry

  Part II, 2016

  For $S \subset \mathbf{R}^{3}$ a smooth embedded surface, define what is meant by a geodesic curve on $S$. Show that any geodesic curve $\gamma(t)$ has constant speed $|\dot{\gamma}(t)|$.

  For any point $P \in S$, show that there is a parametrization $\phi: U \rightarrow V$ of some open neighbourhood $V$ of $P$ in $S$, with $U \subset \mathbf{R}^{2}$ having coordinates $(u, v)$, for which the first fundamental form is

  $$d u^{2}+G(u, v) d v^{2}$$

  for some strictly positive smooth function $G$ on $U$. State a formula for the Gaussian curvature $K$ of $S$ in $V$ in terms of $G$. If $K \equiv 0$ on $V$, show that $G$ is a function of $v$ only, and that we may reparametrize so that the metric is locally of the form $d u^{2}+d w^{2}$, for appropriate local coordinates $(u, w)$.

  [You may assume that for any $P \in S$ and nonzero $\xi \in T_{P} S$, there exists (for some $\epsilon>0)$ a unique geodesic $\gamma:(-\epsilon, \epsilon) \rightarrow S$ with $\gamma(0)=P$ and $\dot{\gamma}(0)=\xi$, and that such geodesics depend smoothly on the initial conditions $P$ and $\xi .]$

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• Paper 2, Section II, G

  Differential Geometry

  Part II, 2016

  If an embedded surface $S \subset \mathbf{R}^{3}$ contains a line $L$, show that the Gaussian curvature is non-positive at each point of $L$. Give an example where the Gaussian curvature is zero at each point of $L$.

  Consider the helicoid $S$ given as the image of $\mathbf{R}^{2}$ in $\mathbf{R}^{3}$ under the map

  $$\phi(u, v)=(\sinh v \cos u, \sinh v \sin u, u) .$$

  What is the image of the corresponding Gauss map? Show that the Gaussian curvature at a point $\phi(u, v) \in S$ is given by $-1 / \cosh ^{4} v$, and hence is strictly negative everywhere. Show moreover that there is a line in $S$ passing through any point of $S$.

  [General results concerning the first and second fundamental forms on an oriented embedded surface $S \subset \mathbf{R}^{3}$ and the Gauss map may be used without proof in this question.]

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• Paper 1, Section II, G

  Differential Geometry

  Part II, 2016

  Define what is meant by the regular values and critical values of a smooth map $f: X \rightarrow Y$ of manifolds. State the Preimage Theorem and Sard's Theorem.

  Suppose now that $\operatorname{dim} X=\operatorname{dim} Y$. If $X$ is compact, prove that the set of regular values is open in $Y$, but show that this may not be the case if $X$ is non-compact.

  Construct an example with $\operatorname{dim} X=\operatorname{dim} Y$ and $X$ compact for which the set of critical values is not a submanifold of $Y$.

  [Hint: You may find it helpful to consider the case $X=S^{1}$ and $Y=\mathbf{R}$. Properties of bump functions and the function $e^{-1 / x^{2}}$ may be assumed in this question.]

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• Paper 3, Section II, E

  Dynamical Systems

  Part II, 2016

  Consider the dependence of the system

  $$\begin{aligned} \dot{x} &=\left(a-x^{2}\right)\left(a^{2}-y\right) \\ \dot{y} &=x-y \end{aligned}$$

  on the parameter $a$. Find the fixed points and plot their location in the $(a, x)$-plane. Hence, or otherwise, deduce that there are bifurcations at $a=0$ and $a=1$.

  Investigate the bifurcation at $a=1$ by making the substitutions $u=x-1, v=y-1$ and $\mu=a-1$. Find the extended centre manifold in the form $v(u, \mu)$ correct to second order. Find the evolution equation on the extended centre manifold to second order, and determine the stability of its fixed points.

  Use a plot to show which branches of fixed points in the $(a, x)$-plane are stable and which are unstable, and state, without calculation, the type of bifurcation at $a=0$. Hence sketch the structure of the $(x, y)$ phase plane very close to the origin for $|a| \ll 1$ in the cases (i) $a<0$ and (ii) $a>0$.

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• Paper 1, Section II, E

  Dynamical Systems

  Part II, 2016

  Consider the dynamical system

  $$\begin{aligned} &\dot{x}=x(y-a), \\ &\dot{y}=1-x-y^{2}, \end{aligned}$$

  where $-1<a<1$. Find and classify the fixed points of the system.

  Use Dulac's criterion with a weighting function of the form $\phi=x^{p}$ and a suitable choice of $p$ to show that there are no periodic orbits for $a \neq 0$. For the case $a=0$ use the same weighting function to find a function $V(x, y)$ which is constant on trajectories. [Hint: $\phi \dot{\mathbf{x}}$ is Hamiltonian.]

  Calculate the stable manifold at $(0,-1)$ correct to quadratic order in $x$.

  Sketch the phase plane for the cases (i) $a=0$ and (ii) $a=\frac{1}{2}$.

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• Paper 4, Section II, E

  Dynamical Systems

  Part II, 2016

  Consider the map defined on $\mathbb{R}$ by

  $$F(x)= \begin{cases}3 x & x \leqslant \frac{1}{2} \\ 3(1-x) & x \geqslant \frac{1}{2}\end{cases}$$

  and let $I$ be the open interval $(0,1)$. Explain what it means for $F$ to have a horseshoe on $I$ by identifying the relevant intervals in the definition.

  Let $\Lambda=\left\{x: F^{n}(x) \in I, \forall n \geqslant 0\right\}$. Show that $F(\Lambda)=\Lambda$.

  Find the sets $\Lambda_{1}=\{x: F(x) \in I\}$ and $\Lambda_{2}=\left\{x: F^{2}(x) \in I\right\}$.

  Consider the ternary (base-3) representation $x=0 \cdot x_{1} x_{2} x_{3} \ldots$ of numbers in $I$. Show that

  $$F\left(0 \cdot x_{1} x_{2} x_{3} \ldots\right)=\left\{\begin{array}{ll} x_{1} \cdot x_{2} x_{3} x_{4} \ldots & x \leqslant \frac{1}{2} \\ \sigma\left(x_{1}\right) \cdot \sigma\left(x_{2}\right) \sigma\left(x_{3}\right) \sigma\left(x_{4}\right) \ldots & x \geqslant \frac{1}{2} \end{array},\right.$$

  where the function $\sigma\left(x_{i}\right)$ of the ternary digits should be identified. What is the ternary representation of the non-zero fixed point? What do the ternary representations of elements of $\Lambda$ have in common?

  Show that $F$ has sensitive dependence on initial conditions on $\Lambda$, that $F$ is topologically transitive on $\Lambda$, and that periodic points are dense in $\Lambda$. [Hint: You may assume that $F^{n}\left(0 \cdot x_{1} \ldots x_{n-1} 0 x_{n+1} x_{n+2} \ldots\right)=0 \cdot x_{n+1} x_{n+2} \ldots$ for $x \in \Lambda$.]

  Briefly state the relevance of this example to the relationship between Glendinning's and Devaney's definitions of chaos.

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• Paper 2, Section II, E

  Dynamical Systems

  Part II, 2016

  Consider the nonlinear oscillator

  $$\begin{aligned} \dot{x} &=y-\mu x\left(\frac{1}{2}|x|-1\right), \\ \dot{y} &=-x \end{aligned}$$

  (a) Use the Hamiltonian for $\mu=0$ to find a constraint on the size of the domain of stability of the origin when $\mu<0$.

  (b) Assume that given $\mu>0$ there exists an $R$ such that all trajectories eventually remain within the region $|\mathbf{x}| \leqslant R$. Show that there must be a limit cycle, stating carefully any result that you use. [You need not show that there is only one periodic orbit.]

  (c) Use the energy-balance method to find the approximate amplitude of the limit cycle for $0<\mu \ll 1$.

  (d) Find the approximate shape of the limit cycle for $\mu \gg 1$, and calculate the leading-order approximation to its period.

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• Paper 1, Section II, E

  Electrodynamics

  Part II, 2016

  A point particle of charge $q$ and mass $m$ moves in an electromagnetic field with 4 -vector potential $A_{\mu}(x)$, where $x^{\mu}$ is position in spacetime. Consider the action

  $$S=-m c \int\left(-\eta_{\mu \nu} \frac{d x^{\mu}}{d \lambda} \frac{d x^{\nu}}{d \lambda}\right)^{1 / 2} d \lambda+q \int A_{\mu} \frac{d x^{\mu}}{d \lambda} d \lambda$$

  where $\lambda$ is an arbitrary parameter along the particle's worldline and $\eta_{\mu \nu}=\operatorname{diag}(-1,+1,+1,+1)$ is the Minkowski metric.

  (a) By varying the action with respect to $x^{\mu}(\lambda)$, with fixed endpoints, obtain the equation of motion

  $$m \frac{d u^{\mu}}{d \tau}=q F_{\nu}^{\mu} u^{\nu} \text {, }$$

  where $\tau$ is the proper time, $u^{\mu}=d x^{\mu} / d \tau$ is the velocity 4-vector, and $F_{\mu \nu}=\partial_{\mu} A_{\nu}-\partial_{\nu} A_{\mu}$ is the field strength tensor.

  (b) This particle moves in the field generated by a second point charge $Q$ that is held at rest at the origin of some inertial frame. By choosing a suitable expression for $A_{\mu}$ and expressing the first particle's spatial position in spherical polar coordinates $(r, \theta, \phi)$, show from the action $(*)$ that

  $$\begin{aligned} \mathcal{E} & \equiv \dot{t}-\Gamma / r \\ \ell c & \equiv r^{2} \dot{\phi} \sin ^{2} \theta \end{aligned}$$

  are constants, where $\Gamma=-q Q /\left(4 \pi \epsilon_{0} m c^{2}\right)$ and overdots denote differentiation with respect to $\tau$.

  (c) Show that when the motion is in the plane $\theta=\pi / 2$,

  $$\mathcal{E}+\frac{\Gamma}{r}=\sqrt{1+\frac{\dot{r}^{2}}{c^{2}}+\frac{\ell^{2}}{r^{2}}}$$

  Hence show that the particle's orbit is bounded if $\mathcal{E}<1$, and that the particle can reach the origin in finite proper time if $\Gamma>|\ell|$.

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• Paper 3, Section II, E

  Electrodynamics

  Part II, 2016

  The current density in an antenna lying along the $z$-axis takes the form

  $$\mathbf{J}(t, \mathbf{x})=\left\{\begin{array}{ll} \hat{\mathbf{z}} I_{0} \sin (k d-k|z|) e^{-i \omega t} \delta(x) \delta(y) & |z| \leqslant d \\ \mathbf{0} & |z|>d \end{array},\right.$$

  where $I_{0}$ is a constant and $\omega=c k$. Show that at distances $r=|\mathbf{x}|$ for which both $r \gg d$ and $r \gg k d^{2} /(2 \pi)$, the retarded vector potential in Lorenz gauge is

  $$\mathbf{A}(t, \mathbf{x}) \approx \hat{\mathbf{z}} \frac{\mu_{0} I_{0}}{4 \pi r} e^{-i \omega(t-r / c)} \int_{-d}^{d} \sin \left(k d-k\left|z^{\prime}\right|\right) e^{-i k z^{\prime} \cos \theta} d z^{\prime}$$

  where $\cos \theta=\hat{\mathbf{r}} \cdot \hat{\mathbf{z}}$ and $\hat{\mathbf{r}}=\mathbf{x} /|\mathbf{x}|$. Evaluate the integral to show that

  $$\mathbf{A}(t, \mathbf{x}) \approx \hat{\mathbf{z}} \frac{\mu_{0} I_{0}}{2 \pi k r}\left(\frac{\cos (k d \cos \theta)-\cos (k d)}{\sin ^{2} \theta}\right) e^{-i \omega(t-r / c)}$$

  In the far-field, where $k r \gg 1$, the electric and magnetic fields are given by

  $$\begin{aligned} &\mathbf{E}(t, \mathbf{x}) \approx-i \omega \hat{\mathbf{r}} \times[\hat{\mathbf{r}} \times \mathbf{A}(t, \mathbf{x})] \\ &\mathbf{B}(t, \mathbf{x}) \approx i k \hat{\mathbf{r}} \times \mathbf{A}(t, \mathbf{x}) \end{aligned}$$

  By calculating the Poynting vector, show that the time-averaged power radiated per unit solid angle is

  $$\frac{d \mathcal{P}}{d \Omega}=\frac{c \mu_{0} I_{0}^{2}}{8 \pi^{2}}\left(\frac{\cos (k d \cos \theta)-\cos (k d)}{\sin \theta}\right)^{2}$$

  [You may assume that in Lorenz gauge, the retarded potential due to a localised current distribution is

  $$\mathbf{A}(t, \mathbf{x})=\frac{\mu_{0}}{4 \pi} \int \frac{\mathbf{J}\left(t_{\mathrm{ret}}, \mathbf{x}^{\prime}\right)}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|} d^{3} \mathbf{x}^{\prime}$$

  where the retarded time $\left.t_{\text {ret }}=t-\left|\mathbf{x}-\mathbf{x}^{\prime}\right| / c .\right]$

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• Paper 4, Section II, E

  Electrodynamics

  Part II, 2016

  (a) A uniform, isotropic dielectric medium occupies the half-space $z>0$. The region $z<0$ is in vacuum. State the boundary conditions that should be imposed on $\mathbf{E}, \mathbf{D}, \mathbf{B}$ and $\mathbf{H}$ at $z=0$.

  (b) A linearly polarized electromagnetic plane wave, with magnetic field in the $(x, y)$-plane, is incident on the dielectric from $z<0$. The wavevector $\mathbf{k}$ makes an acute angle $\theta_{I}$ to the normal $\hat{\mathbf{z}}$. If the dielectric has frequency-independent relative permittivity $\epsilon_{r}$, show that the fraction of the incident power that is reflected is

  $$\mathcal{R}=\left(\frac{n \cos \theta_{I}-\cos \theta_{T}}{n \cos \theta_{I}+\cos \theta_{T}}\right)^{2}$$

  where $n=\sqrt{\epsilon_{r}}$, and the angle $\theta_{T}$ should be specified. [You should ignore any magnetic response of the dielectric.]

  (c) Now suppose that the dielectric moves at speed $\beta c$ along the $x$-axis, the incident angle $\theta_{I}=0$, and the magnetic field of the incident radiation is along the $y$-direction. Show that the reflected radiation propagates normal to the surface $z=0$, has the same frequency as the incident radiation, and has magnetic field also along the $y$-direction. [Hint: You may assume that under a standard Lorentz boost with speed $v=\beta c$ along the $x$-direction, the electric and magnetic field components transform as

  $$\left(\begin{array}{c} E_{x}^{\prime} \\ E_{y}^{\prime} \\ E_{z}^{\prime} \end{array}\right)=\left(\begin{array}{c} E_{x} \\ \gamma\left(E_{y}-v B_{z}\right) \\ \gamma\left(E_{z}+v B_{y}\right) \end{array}\right) \quad \text { and } \quad\left(\begin{array}{c} B_{x}^{\prime} \\ B_{y}^{\prime} \\ B_{z}^{\prime} \end{array}\right)=\left(\begin{array}{c} B_{x} \\ \gamma\left(B_{y}+v E_{z} / c^{2}\right) \\ \gamma\left(B_{z}-v E_{y} / c^{2}\right) \end{array}\right)$$

  where $\gamma=\left(1-\beta^{2}\right)^{-1 / 2}$.]

  (d) Show that the fraction of the incident power reflected from the moving dielectric

  $$\mathcal{R}_{\beta}=\left(\frac{n / \gamma-\sqrt{1-\beta^{2} / n^{2}}}{n / \gamma+\sqrt{1-\beta^{2} / n^{2}}}\right)^{2}$$

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• Paper 4 , Section II, B

  Fluid Dynamics II

  Part II, 2016

  A thin layer of fluid of viscosity $\mu$ occupies the gap between a rigid flat plate at $y=0$ and a flexible no-slip boundary at $y=h(x, t)$. The flat plate moves with constant velocity $U \mathbf{e}_{x}$ and the flexible boundary moves with no component of velocity in the $x$-direction.

  State the two-dimensional lubrication equations governing the dynamics of the thin layer of fluid. Given a pressure gradient $\mathrm{d} p / \mathrm{d} x$, solve for the velocity profile $u(x, y, t)$ in the fluid and calculate the flux $q(x, t)$. Deduce that the pressure gradient satisfies

  $$\frac{\partial}{\partial x}\left(\frac{h^{3}}{12 \mu} \frac{\mathrm{d} p}{\mathrm{~d} x}\right)=\frac{\partial h}{\partial t}+\frac{U}{2} \frac{\partial h}{\partial x}$$

  The shape of the flexible boundary is a periodic travelling wave, i.e. $h(x, t)=$ $h(x-c t)$ and $h(\xi+L)=h(\xi)$, where $c$ and $L$ are constants. There is no applied average pressure gradient, so the pressure is also periodic with $p(\xi+L)=p(\xi)$. Show that

  $$\frac{\mathrm{d} p}{\mathrm{~d} x}=6 \mu(U-2 c)\left(\frac{1}{h^{2}}-\frac{\left\langle h^{-2}\right\rangle}{\left\langle h^{-3}\right\rangle} \frac{1}{h^{3}}\right)$$

  where $\langle\ldots\rangle=\frac{1}{L} \int_{0}^{L} \ldots \mathrm{d} x$ denotes the average over a period. Calculate the shear stress $\sigma_{x y}$ on the plate.

  The speed $U$ is such that there is no need to apply an external tangential force to the plate in order to maintain its motion. Show that

  $$U=6 c \frac{\left\langle h^{-2}\right\rangle\left\langle h^{-2}\right\rangle-\left\langle h^{-1}\right\rangle\left\langle h^{-3}\right\rangle}{3\left\langle h^{-2}\right\rangle\left\langle h^{-2}\right\rangle-4\left\langle h^{-1}\right\rangle\left\langle h^{-3}\right\rangle} .$$

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• Paper 3, Section II, B

  Fluid Dynamics II

  Part II, 2016

  A cylindrical pipe of radius $a$ and length $L \gg a$ contains two viscous fluids arranged axisymmetrically with fluid 1 of viscosity $\mu_{1}$ occupying the central region $r<\beta a$, where $0<\beta<1$, and fluid 2 of viscosity $\mu_{2}$ occupying the surrounding annular region $\beta a<r<a$. The flow in each fluid is assumed to be steady and unidirectional, with velocities $u_{1}(r) \mathbf{e}_{z}$ and $u_{2}(r) \mathbf{e}_{z}$ respectively, with respect to cylindrical coordinates $(r, \theta, z)$ aligned with the pipe. A fixed pressure drop $\Delta p$ is applied between the ends of the pipe.

  Starting from the Navier-Stokes equations, derive the equations satisfied by $u_{1}(r)$ and $u_{2}(r)$, and state all the boundary conditions. Show that the pressure gradient is constant.

  Solve for the velocity profile in each fluid and calculate the corresponding flow rates, $Q_{1}$ and $Q_{2}$.

  Derive the relationship between $\beta$ and $\mu_{2} / \mu_{1}$ that yields the same flow rate in each fluid. Comment on the behaviour of $\beta$ in the limits $\mu_{2} / \mu_{1} \gg 1$ and $\mu_{2} / \mu_{1} \ll 1$, illustrating your comment by sketching the flow profiles.

  $[$ Hint: In cylindrical coordinates $(r, \theta, z)$,

  $$\nabla^{2} u=\frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial u}{\partial r}\right)+\frac{1}{r^{2}} \frac{\partial^{2} u}{\partial \theta^{2}}+\frac{\partial^{2} u}{\partial z^{2}}, \quad e_{r z}=\frac{1}{2}\left(\frac{\partial u_{r}}{\partial z}+\frac{\partial u_{z}}{\partial r}\right)$$

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• Paper 2, Section II, B

  Fluid Dynamics II

  Part II, 2016

  For a two-dimensional flow in plane polar coordinates $(r, \theta)$, state the relationship between the streamfunction $\psi(r, \theta)$ and the flow components $u_{r}$ and $u_{\theta}$. Show that the vorticity $\omega$ is given by $\omega=-\nabla^{2} \psi$, and deduce that the streamfunction for a steady two-dimensional Stokes flow satisfies the biharmonic equation

  $$\nabla^{4} \psi=0$$

  A rigid stationary circular disk of radius $a$ occupies the region $r \leqslant a$. The flow far from the disk tends to a steady straining flow $\mathbf{u}_{\infty}=(-E x, E y)$, where $E$ is a constant. Inertial forces may be neglected. Calculate the streamfunction, $\psi_{\infty}(r, \theta)$, for the far-field flow.

  By making an appropriate assumption about its dependence on $\theta$, find the streamfunction $\psi$ for the flow around the disk, and deduce the flow components, $u_{r}(r, \theta)$ and $u_{\theta}(r, \theta)$.

  Calculate the tangential surface stress, $\sigma_{r \theta}$, acting on the boundary of the disk.

  $[$ Hints: In plane polar coordinates $(r, \theta)$,

  $$\begin{gathered} \boldsymbol{\nabla} \cdot \mathbf{u}=\frac{1}{r} \frac{\partial\left(r u_{r}\right)}{\partial r}+\frac{1}{r} \frac{\partial u_{\theta}}{\partial \theta}, \quad \omega=\frac{1}{r} \frac{\partial\left(r u_{\theta}\right)}{\partial r}-\frac{1}{r} \frac{\partial u_{r}}{\partial \theta} \\ \nabla^{2} V=\frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial V}{\partial r}\right)+\frac{1}{r^{2}} \frac{\partial^{2} V}{\partial \theta^{2}}, \quad e_{r \theta}=\frac{1}{2}\left(r \frac{\partial}{\partial r}\left(\frac{u_{\theta}}{r}\right)+\frac{1}{r} \frac{\partial u_{r}}{\partial \theta}\right) \end{gathered}$$

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• Paper 1, Section II, B

  Fluid Dynamics II

  Part II, 2016

  State the vorticity equation and interpret the meaning of each term.

  A planar vortex sheet is diffusing in the presence of a perpendicular straining flow. The flow is everywhere of the form $\mathbf{u}=(U(y, t),-E y, E z)$, where $U \rightarrow \pm U_{0}$ as $y \rightarrow \pm \infty$, and $U_{0}$ and $E>0$ are constants. Show that the vorticity has the form $\boldsymbol{\omega}=\omega(y, t) \mathbf{e}_{z}$, and obtain a scalar equation describing the evolution of $\omega$.

  Explain physically why the solution approaches a steady state in which the vorticity is concentrated near $y=0$. Use scaling to estimate the thickness $\delta$ of the steady vorticity layer as a function of $E$ and the kinematic viscosity $\nu$.

  Determine the steady vorticity profile, $\omega(y)$, and the steady velocity profile, $U(y)$.

  $\left[\right.$ Hint: $\left.\quad \operatorname{erf}(x)=\frac{2}{\sqrt{\pi}} \int_{0}^{x} e^{-u^{2}} \mathrm{~d} u .\right]$

  State, with a brief physical justification, why you might expect this steady flow to be unstable to long-wavelength perturbations, defining what you mean by long.

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• Paper 4, Section I, 7A

  Further Complex Methods

  Part II, 2016

  Consider the equation for $w(z)$ :

  $$w^{\prime \prime}+p(z) w^{\prime}+q(z) w=0 .$$

  State necessary and sufficient conditions on $p(z)$ and $q(z)$ for $z=0$ to be (i) an ordinary point or (ii) a regular singular point. Derive the corresponding conditions for the point $z=\infty$.

  Determine the most general equation of the form $(*)$ that has regular singular points at $z=0$ and $z=\infty$, with all other points being ordinary.

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• Paper 3, Section I, A

  Further Complex Methods

  Part II, 2016

  The functions $f(x)$ and $g(x)$ have Laplace transforms $F(p)$ and $G(p)$ respectively, and $f(x)=g(x)=0$ for $x \leqslant 0$. The convolution $h(x)$ of $f(x)$ and $g(x)$ is defined by

  $$h(x)=\int_{0}^{x} f(y) g(x-y) d y \quad \text { for } \quad x>0 \quad \text { and } \quad h(x)=0 \quad \text { for } \quad x \leqslant 0$$

  Express the Laplace transform $H(p)$ of $h(x)$ in terms of $F(p)$ and $G(p)$.

  Now suppose that $f(x)=x^{\alpha}$ and $g(x)=x^{\beta}$ for $x>0$, where $\alpha, \beta>-1$. Find expressions for $F(p)$ and $G(p)$ by using a standard integral formula for the Gamma function. Find an expression for $h(x)$ by using a standard integral formula for the Beta function. Hence deduce that

  $$\frac{\Gamma(z) \Gamma(w)}{\Gamma(z+w)}=\mathrm{B}(z, w)$$

  for all $\operatorname{Re}(z)>0, \operatorname{Re}(w)>0$.

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• Paper 1, Section I, A

  Further Complex Methods

  Part II, 2016

  Evaluate the integral

  $$f(p)=\mathcal{P} \int_{-\infty}^{\infty} d x \frac{e^{i p x}}{x^{4}-1}$$

  where $p$ is a real number, for (i) $p>0$ and (ii) $p<0$.

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• Paper 2, Section I, A

  Further Complex Methods

  Part II, 2016

  The Euler product formula for the Gamma function is

  $$\Gamma(z)=\lim _{n \rightarrow \infty} \frac{n ! n^{z}}{z(z+1) \ldots(z+n)}$$

  Use this to show that

  $$\frac{\Gamma(2 z)}{2^{2 z} \Gamma(z) \Gamma\left(z+\frac{1}{2}\right)}=c,$$

  where $c$ is a constant, independent of $z$. Find the value of $c$.

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• Paper 2, Section II, A

  Further Complex Methods

  Part II, 2016

  The Hurwitz zeta function $\zeta_{\mathrm{H}}(s, q)$ is defined for $\operatorname{Re}(q)>0$ by

  $$\zeta_{\mathrm{H}}(s, q)=\sum_{n=0}^{\infty} \frac{1}{(q+n)^{s}}$$

  State without proof the complex values of $s$ for which this series converges.

  Consider the integral

  $$I(s, q)=\frac{\Gamma(1-s)}{2 \pi i} \int_{\mathcal{C}} d z \frac{z^{s-1} e^{q z}}{1-e^{z}}$$

  where $\mathcal{C}$ is the Hankel contour. Show that $I(s, q)$ provides an analytic continuation of the Hurwitz zeta function for all $s \neq 1$. Include in your account a careful discussion of removable singularities. [Hint: $\Gamma(s) \Gamma(1-s)=\pi / \sin (\pi s)$.]

  Show that $I(s, q)$ has a simple pole at $s=1$ and find its residue.

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• Paper 1, Section II, A

  Further Complex Methods

  Part II, 2016

  (a) Legendre's equation for $w(z)$ is

  $$\left(z^{2}-1\right) w^{\prime \prime}+2 z w^{\prime}-\ell(\ell+1) w=0, \quad \text { where } \quad \ell=0,1,2, \ldots$$

  Let $\mathcal{C}$ be a closed contour. Show by direct substitution that for $z$ within $\mathcal{C}$

  $$\int_{\mathcal{C}} d t \frac{\left(t^{2}-1\right)^{\ell}}{(t-z)^{\ell+1}}$$

  is a non-trivial solution of Legendre's equation.

  (b) Now consider

  $$Q_{\nu}(z)=\frac{1}{4 i \sin \nu \pi} \int_{\mathcal{C}^{\prime}} d t \frac{\left(t^{2}-1\right)^{\nu}}{(t-z)^{\nu+1}}$$

  for real $\nu>-1$ and $\nu \neq 0,1,2, \ldots$. The closed contour $\mathcal{C}^{\prime}$ is defined to start at the origin, wind around $t=1$ in a counter-clockwise direction, then wind around $t=-1$ in a clockwise direction, then return to the origin, without encircling the point $z$. Assuming that $z$ does not lie on the real interval $-1 \leqslant x \leqslant 1$, show by deforming $\mathcal{C}^{\prime}$ onto this interval that functions $Q_{\ell}(z)$ may be defined as limits of $Q_{\nu}(z)$ with $\nu \rightarrow \ell=0,1,2, \ldots$.

  Find an explicit expression for $Q_{0}(z)$ and verify that it satisfies Legendre's equation with $\ell=0$.

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• Paper 2, Section II, H

  Galois Theory

  Part II, 2016

  (a) Let $K \subseteq L$ be a finite separable field extension. Show that there exist only finitely many intermediate fields $K \subseteq F \subseteq L$.

  (b) Define what is meant by a normal extension. Is $\mathbb{Q} \subseteq \mathbb{Q}(\sqrt{1+\sqrt{7}})$ a normal extension? Justify your answer.

  (c) Prove Artin's lemma, which states: if $K \subseteq L$ is a field extension, $H$ is a finite subgroup of $\operatorname{Aut}_{K}(L)$, and $F:=L^{H}$ is the fixed field of $H$, then $F \subseteq L$ is a Galois extension with $\operatorname{Gal}(L / F)=H$.

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• Paper 3, Section II, H

  Galois Theory

  Part II, 2016

  (a) Let $L$ be the 13 th cyclotomic extension of $\mathbb{Q}$, and let $\mu$ be a 13 th primitive root of unity. What is the minimal polynomial of $\mu$ over $\mathbb{Q}$ ? What is the Galois group $\operatorname{Gal}(L / \mathbb{Q})$ ? Put $\lambda=\mu+\frac{1}{\mu}$. Show that $\mathbb{Q} \subseteq \mathbb{Q}(\lambda)$ is a Galois extension and find $\operatorname{Gal}(\mathbb{Q}(\lambda) / \mathbb{Q})$.

  (b) Define what is meant by a Kummer extension. Let $K$ be a field of characteristic zero and let $L$ be the $n$th cyclotomic extension of $K$. Show that there is a sequence of Kummer extensions $K=F_{1} \subseteq F_{2} \subseteq \cdots \subseteq F_{r}$ such that $L$ is contained in $F_{r}$.

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• Paper 1, Section II, H

  Galois Theory

  Part II, 2016

  (a) Prove that if $K$ is a field and $f \in K[t]$, then there exists a splitting field $L$ of $f$ over $K$. [You do not need to show uniqueness of $L$.]

  (b) Let $K_{1}$ and $K_{2}$ be algebraically closed fields of the same characteristic. Show that either $K_{1}$ is isomorphic to a subfield of $K_{2}$ or $K_{2}$ is isomorphic to a subfield of $K_{1}$. [For subfields $F_{i}$ of $K_{1}$ and field homomorphisms $\psi_{i}: F_{i} \rightarrow K_{2}$ with $i=1$, 2, we say $\left(F_{1}, \psi_{1}\right) \leqslant\left(F_{2}, \psi_{2}\right)$ if $F_{1}$ is a subfield of $F_{2}$ and $\left.\psi_{2}\right|_{F_{1}}=\psi_{1}$. You may assume the existence of a maximal pair $(F, \psi)$ with respect to the partial order just defined.]

  (c) Give an example of a finite field extension $K \subseteq L$ such that there exist $\alpha, \beta \in L \backslash K$ where $\alpha$ is separable over $K$ but $\beta$ is not separable over $K .$

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• Paper 4, Section II, H

  Galois Theory

  Part II, 2016

  (a) Let $f=t^{5}-9 t+3 \in \mathbb{Q}[t]$ and let $L$ be the splitting field of $f$ over $\mathbb{Q}$. Show that $\operatorname{Gal}(L / \mathbb{Q})$ is isomorphic to $S_{5}$. Let $\alpha$ be a root of $f$. Show that $\mathbb{Q} \subseteq \mathbb{Q}(\alpha)$ is neither a radical extension nor a solvable extension.

  (b) Let $f=t^{26}+2$ and let $L$ be the splitting field of $f$ over $\mathbb{Q}$. Is it true that $\operatorname{Gal}(L / \mathbb{Q})$ has an element of order 29 ? Justify your answer. Using reduction mod $p$ techniques, or otherwise, show that $\operatorname{Gal}(L / \mathbb{Q})$ has an element of order 3 .

  [Standard results from the course may be used provided they are clearly stated.]

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• Paper 4, Section II, D

  General Relativity

  Part II, 2016

  A spherically symmetric static spacetime has metric

  $$d s^{2}=-\left(1+r^{2} / b^{2}\right) d t^{2}+\frac{d r^{2}}{1+r^{2} / b^{2}}+r^{2}\left(d \theta^{2}+\sin ^{2} \theta d \phi^{2}\right)$$

  where $-\infty<t<\infty, r \geqslant 0, b$ is a positive constant, and units such that $c=1$ are used.

  (a) Explain why a time-like geodesic may be assumed, without loss of generality, to lie in the equatorial plane $\theta=\pi / 2$. For such a geodesic, show that the quantities

  $$E=\left(1+r^{2} / b^{2}\right) \dot{t} \quad \text { and } \quad h=r^{2} \dot{\phi}$$

  are constants of the motion, where a dot denotes differentiation with respect to proper time, $\tau$. Hence find a first-order differential equation for $r(\tau)$.

  (b) Consider a massive particle fired from the origin, $r=0$. Show that the particle will return to the origin and find the proper time taken.

  (c) Show that circular orbits $r=a$ are possible for any $a>0$ and determine whether such orbits are stable. Show that on any such orbit a clock measures coordinate time.

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• Paper 1, Section II, D

  General Relativity

  Part II, 2016

  Consider a family of geodesics with $s$ an affine parameter and $V^{a}$ the tangent vector on each curve. The equation of geodesic deviation for a vector field $W^{a}$ is

  $$\frac{D^{2} W^{a}}{D s^{2}}=R_{b c d}^{a} V^{b} V^{c} W^{d}$$

  where $\frac{D}{D s}$ denotes the directional covariant derivative $V^{b} \nabla_{b}$.

  (i) Show that if

  $$V^{b} \frac{\partial W^{a}}{\partial x^{b}}=W^{b} \frac{\partial V^{a}}{\partial x^{b}}$$

  then $W^{a}$ satisfies $(*)$.

  (ii) Show that $V^{a}$ and $s V^{a}$ satisfy $(*)$.

  (iii) Show that if $W^{a}$ is a Killing vector field, meaning that $\nabla_{b} W_{a}+\nabla_{a} W_{b}=0$, then $W^{a}$ satisfies $(*)$.

  (iv) Show that if $W^{a}=w U^{a}$ satisfies $(*)$, where $w$ is a scalar field and $U^{a}$ is a time-like unit vector field, then

  $$\begin{gathered} \frac{d^{2} w}{d s^{2}}=\left(\Omega^{2}-K\right) w \\ -\frac{D U^{a}}{D s} \frac{D U_{a}}{D s} \quad \text { and } \quad K=R_{a b c d} U^{a} V^{b} V^{c} U^{d} \end{gathered}$$

  $$\begin{aligned} & \text { where } \quad \Omega^{2}=-\frac{D U^{a}}{D s} \frac{D U_{a}}{D s} \text { and } \quad K=R_{a b c d} U^{a} V^{b} V^{c} U^{d} \text {. } \end{aligned}$$

  [You may use: $\nabla_{b} \nabla_{c} X^{a}-\nabla_{c} \nabla_{b} X^{a}=R_{d b c}^{a} X^{d}$ for any vector field $\left.X^{a} .\right]$

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• Paper 2, Section II, D

  General Relativity

  Part II, 2016

  The Kasner (vacuum) cosmological model is defined by the line element

  $$d s^{2}=-c^{2} d t^{2}+t^{2 p_{1}} d x^{2}+t^{2 p_{2}} d y^{2}+t^{2 p_{3}} d z^{2} \quad \text { with } \quad t>0$$

  where $p_{1}, p_{2}, p_{3}$ are constants with $p_{1}+p_{2}+p_{3}=p_{1}^{2}+p_{2}^{2}+p_{3}^{2}=1$ and $0<p_{1}<1$. Show that $p_{2} p_{3}<0$.

  Write down four equations that determine the null geodesics of the Kasner model.

  If $k^{a}$ is the tangent vector to the trajectory of a photon and $u^{a}$ is the four-velocity of a comoving observer (i.e., an observer at rest in the $(t, x, y, z)$ coordinate system above), what is the physical interpretation of $k_{a} u^{a}$ ?

  Let $O$ be a comoving observer at the origin, $x=y=z=0$, and let $S$ be a comoving source of photons located on one of the spatial coordinate axes.

  (i) Show that photons emitted by $S$ and observed by $O$ can be either redshifted or blue-shifted, depending on the location of $S$.

  (ii) Given any fixed time $t=T$, show that there are locations for $S$ on each coordinate axis from which no photons reach $O$ for $t \leqslant T$.

  Now suppose that $p_{1}=1$ and $p_{2}=p_{3}=0$. Does the property in (ii) still hold?

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• Paper 3, Section II, D

  General Relativity

  Part II, 2016

  For a spacetime that is nearly flat, the metric $g_{a b}$ can be expressed in the form

  $$g_{a b}=\eta_{a b}+h_{a b}$$

  where $\eta_{a b}$ is a flat metric (not necessarily diagonal) with constant components, and the components of $h_{a b}$ and their derivatives are small. Show that

  $$2 R_{b d} \approx h_{d}^{a}, b a+h_{b}^{a}, d a-h_{a, b d}^{a}-h_{b d, a c} \eta^{a c}$$

  where indices are raised and lowered using $\eta_{a b}$.

  [You may assume that $\left.R_{b c d}^{a}=\Gamma_{b d, c}^{a}-\Gamma_{b c, d}^{a}+\Gamma_{c e}^{a} \Gamma_{d b}^{e}-\Gamma_{d e}^{a} \Gamma_{c b}^{e} .\right]$

  For the line element

  $$d s^{2}=2 d u d v+d x^{2}+d y^{2}+H(u, x, y) d u^{2},$$

  where $H$ and its derivatives are small, show that the linearised vacuum field equations reduce to $\nabla^{2} H=0$, where $\nabla^{2}$ is the two-dimensional Laplacian operator in $x$ and $y$.

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• Paper 3, Section II, G

  Graph Theory

  Part II, 2016

  Define the chromatic polynomial $p_{G}(t)$ of a graph $G$. Show that if $G$ has $n$ vertices and $m$ edges then

  $$p_{G}(t)=a_{n} t^{n}-a_{n-1} t^{n-1}+a_{n-2} t^{n-2}-\ldots+(-1)^{n} a_{0}$$

  where $a_{n}=1, a_{n-1}=m$ and $a_{i} \geqslant 0$ for all $i$. [You may assume the deletion-contraction relation, provided that you state it clearly.]

  Show that for every graph $G$ (with $n>0$ ) we have $a_{0}=0$. Show also that $a_{1}=0$ if and only if $G$ is disconnected.

  Explain why $t^{4}-2 t^{3}+3 t^{2}-t$ is not the chromatic polynomial of any graph.

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• Paper 2, Section II, G

  Graph Theory

  Part II, 2016

  Define the Turán graph $T_{r}(n)$, where $r$ and $n$ are positive integers with $n \geqslant r$. For which $r$ and $n$ is $T_{r}(n)$ regular? For which $r$ and $n$ does $T_{r}(n)$ contain $T_{4}(8)$ as a subgraph?

  State and prove Turán's theorem.

  Let $x_{1}, \ldots, x_{n}$ be unit vectors in the plane. Prove that the number of pairs $i<j$ for which $x_{i}+x_{j}$ has length less than 1 is at most $\left\lfloor n^{2} / 4\right\rfloor$.

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• Paper 4, Section II, G

  Graph Theory

  Part II, 2016

  State Menger's theorem in both the vertex form and the edge form. Explain briefly how the edge form of Menger's theorem may be deduced from the vertex form.

  (a) Show that if $G$ is 3 -connected then $G$ contains a cycle of even length.

  (b) Let $G$ be a connected graph with all degrees even. Prove that $\lambda(G)$ is even. [Hint: if $S$ is a minimal set of edges whose removal disconnects $G$, let $H$ be a component of $G-S$ and consider the degrees of the vertices of $H$ in the graph $G-S$.] Give an example to show that $\kappa(G)$ can be odd.

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• Paper 1, Section II, 16G

  Graph Theory

  Part II, 2016

  (a) Show that if $G$ is a planar graph then $\chi(G) \leqslant 5$. [You may assume Euler's formula, provided that you state it precisely.]

  (b) (i) Prove that if $G$ is a triangle-free planar graph then $\chi(G) \leqslant 4$.

  (ii) Prove that if $G$ is a planar graph of girth at least 6 then $\chi(G) \leqslant 3$.

  (iii) Does there exist a constant $g$ such that, if $G$ is a planar graph of girth at least $g$, then $\chi(G) \leqslant 2 ?$ Justify your answer.

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• Paper 3, Section II, D

  Integrable Systems

  Part II, 2016

  What is meant by an auto-Bäcklund transformation?

  The sine-Gordon equation in light-cone coordinates is

  $$\frac{\partial^{2} \varphi}{\partial \xi \partial \tau}=\sin \varphi$$

  where $\xi=\frac{1}{2}(x-t), \tau=\frac{1}{2}(x+t)$ and $\varphi$ is to be understood modulo $2 \pi$. Show that the pair of equations

  $$\partial_{\xi}\left(\varphi_{1}-\varphi_{0}\right)=2 \epsilon \sin \left(\frac{\varphi_{1}+\varphi_{0}}{2}\right), \quad \partial_{\tau}\left(\varphi_{1}+\varphi_{0}\right)=\frac{2}{\epsilon} \sin \left(\frac{\varphi_{1}-\varphi_{0}}{2}\right)$$

  constitute an auto-Bäcklund transformation for (1).

  By noting that $\varphi=0$ is a solution to (1), use the transformation (2) to derive the soliton (or 'kink') solution to the sine-Gordon equation. Show that this solution can be expressed as

  $$\varphi(x, t)=4 \arctan \left[\exp \left(\pm \frac{x-c t}{\sqrt{1-c^{2}}}+x_{0}\right)\right]$$

  for appropriate constants $c$ and $x_{0}$.

  [Hint: You may use the fact that $\int \operatorname{cosec} x \mathrm{~d} x=\log \tan (x / 2)+$ const.]

  The following function is a solution to the sine-Gordon equation:

  $$\varphi(x, t)=4 \arctan \left[c \frac{\sinh \left(x / \sqrt{1-c^{2}}\right)}{\cosh \left(c t / \sqrt{1-c^{2}}\right)}\right] \quad(c>0) .$$

  Verify that this represents two solitons travelling towards each other at the same speed by considering $x \pm c t=$ constant and taking an appropriate limit.

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• Paper 1, Section II, D

  Integrable Systems

  Part II, 2016

  What does it mean for an evolution equation $u_{t}=K\left(x, u, u_{x}, \ldots\right)$ to be in Hamiltonian form? Define the associated Poisson bracket.

  An evolution equation $u_{t}=K\left(x, u, u_{x}, \ldots\right)$ is said to be bi-Hamiltonian if it can be written in Hamiltonian form in two distinct ways, i.e.

  $$K=\mathcal{J} \delta H_{0}=\mathcal{E} \delta H_{1}$$

  for Hamiltonian operators $\mathcal{J}, \mathcal{E}$ and functionals $H_{0}, H_{1}$. By considering the sequence $\left\{H_{m}\right\}_{m \geqslant 0}$ defined by the recurrence relation

  $$\mathcal{E} \delta H_{m+1}=\mathcal{J} \delta H_{m}$$

  show that bi-Hamiltonian systems possess infinitely many first integrals in involution. [You may assume that $(*)$ can always be solved for $H_{m+1}$, given $H_{m}$.]

  The Harry Dym equation for the function $u=u(x, t)$ is

  $$u_{t}=\frac{\partial^{3}}{\partial x^{3}}\left(u^{-1 / 2}\right)$$

  This equation can be written in Hamiltonian form $u_{t}=\mathcal{E} \delta H_{1}$ with

  $$\mathcal{E}=2 u \frac{\partial}{\partial x}+u_{x}, \quad H_{1}[u]=\frac{1}{8} \int u^{-5 / 2} u_{x}^{2} \mathrm{~d} x$$

  Show that the Harry Dym equation possesses infinitely many first integrals in involution. [You need not verify the Jacobi identity if your argument involves a Hamiltonian operator.]

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• Paper 2, Section II, D

  Integrable Systems

  Part II, 2016

  What does it mean for $g^{\epsilon}:(x, u) \mapsto(\tilde{x}, \tilde{u})$ to describe a 1-parameter group of transformations? Explain how to compute the vector field

  $$V=\xi(x, u) \frac{\partial}{\partial x}+\eta(x, u) \frac{\partial}{\partial u}$$

  that generates such a 1-parameter group of transformations.

  Suppose now $u=u(x)$. Define the $n$th prolongation, $\mathrm{pr}^{(n)} g^{\epsilon}$, of $g^{\epsilon}$ and the vector field which generates it. If $V$ is defined by $(*)$ show that

  $$\mathrm{pr}^{(n)} V=V+\sum_{k=1}^{n} \eta_{k} \frac{\partial}{\partial u^{(k)}}$$

  where $u^{(k)}=\mathrm{d}^{k} u / \mathrm{d} x^{k}$ and $\eta_{k}$ are functions to be determined.

  The curvature of the curve $u=u(x)$ in the $(x, u)$-plane is given by

  $$\kappa=\frac{u_{x x}}{\left(1+u_{x}^{2}\right)^{3 / 2}}$$

  Rotations in the $(x, u)$-plane are generated by the vector field

  $$W=x \frac{\partial}{\partial u}-u \frac{\partial}{\partial x}$$

  Show that the curvature $\kappa$ at a point along a plane curve is invariant under such rotations. Find two further transformations that leave $\kappa$ invariant.

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• Paper 3, Section II, I

  Linear Analysis

  Part II, 2016

  (a) Define Banach spaces and Euclidean spaces over $\mathbb{R}$. [You may assume the definitions of vector spaces and inner products.]

  (b) Let $X$ be the space of sequences of real numbers with finitely many non-zero entries. Does there exist a norm $\|\cdot\|$ on $X$ such that $(X,\|\cdot\|)$ is a Banach space? Does there exist a norm such that $(X,\|\cdot\|)$ is Euclidean? Justify your answers.

  (c) Let $(X,\|\cdot\|)$ be a normed vector space over $\mathbb{R}$ satisfying the parallelogram law

  $$\|x+y\|^{2}+\|x-y\|^{2}=2\|x\|^{2}+2\|y\|^{2}$$

  for all $x, y \in X$. Show that $\langle x, y\rangle=\frac{1}{4}\left(\|x+y\|^{2}-\|x-y\|^{2}\right)$ is an inner product on $X$. [You may use without proof the fact that the vector space operations $+$ and are continuous with respect to $\|\cdot\|$. To verify the identity $\langle a+b, c\rangle=\langle a, c\rangle+\langle b, c\rangle$, you may find it helpful to consider the parallelogram law for the pairs $(a+c, b),(b+c, a),(a-c, b)$ and $(b-c, a) .]$

  (d) Let $\left(X,\|\cdot\|_{X}\right)$ be an incomplete normed vector space over $\mathbb{R}$ which is not a Euclidean space, and let $\left(X^{*},\|\cdot\|_{X^{*}}\right)$ be its dual space with the dual norm. Is $\left(X^{*},\|\cdot\|_{X^{*}}\right)$ a Banach space? Is it a Euclidean space? Justify your answers.

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• Paper 2, Section II, I

  Linear Analysis

  Part II, 2016

  (a) Let $K$ be a topological space and let $C_{\mathbb{R}}(K)$ denote the normed vector space of bounded continuous real-valued functions on $K$ with the norm $\|f\|_{C_{\mathbb{R}}(K)}=\sup _{x \in K}|f(x)|$. Define the terms uniformly bounded, equicontinuous and relatively compact as applied to subsets $S \subset C_{\mathbb{R}}(K)$.

  (b) The Arzela-Ascoli theorem [which you need not prove] states in particular that if $K$ is compact and $S \subset C_{\mathbb{R}}(K)$ is uniformly bounded and equicontinuous, then $S$ is relatively compact. Show by examples that each of the compactness of $K$, uniform boundedness of $S$, and equicontinuity of $S$ are necessary conditions for this conclusion.

  (c) Let $L$ be a topological space. Assume that there exists a sequence of compact subsets $K_{n}$ of $L$ such that $K_{1} \subset K_{2} \subset K_{3} \subset \cdots \subset L$ and $\bigcup_{n=1}^{\infty} K_{n}=L$. Suppose $S \subset C_{\mathbb{R}}(L)$ is uniformly bounded and equicontinuous and moreover satisfies the condition that, for every $\epsilon>0$, there exists $n \in \mathbb{N}$ such that $|f(x)|<\epsilon$ for every $x \in L \backslash K_{n}$ and for every $f \in S$. Show that $S$ is relatively compact.

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• Paper 1, Section II, I

  Linear Analysis

  Part II, 2016

  (a) State the closed graph theorem.

  (b) Prove the closed graph theorem assuming the inverse mapping theorem.

  (c) Let $X, Y, Z$ be Banach spaces and $T: X \rightarrow Y, S: Y \rightarrow Z$ be linear maps. Suppose that $S \circ T$ is bounded and $S$ is both bounded and injective. Show that $T$ is bounded.

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• Paper 4, Section II, I

  Linear Analysis

  Part II, 2016

  Let $H$ be a complex Hilbert space.

  (a) Let $T: H \rightarrow H$ be a bounded linear map. Show that the spectrum of $T$ is a subset of $\left\{\lambda \in \mathbb{C}:|\lambda| \leqslant\|T\|_{\mathcal{B}(H)}\right\}$.

  (b) Let $T: H \rightarrow H$ be a bounded self-adjoint linear map. For $\lambda, \mu \in \mathbb{C}$, let $E_{\lambda}:=\{x \in H: T x=\lambda x\}$ and $E_{\mu}:=\{x \in H: T x=\mu x\}$. If $\lambda \neq \mu$, show that $E_{\lambda} \perp E_{\mu}$.

  (c) Let $T: H \rightarrow H$ be a compact self-adjoint linear map. For $\lambda \neq 0$, show that $E_{\lambda}:=\{x \in H: T x=\lambda x\}$ is finite-dimensional.

  (d) Let $H_{1} \subset H$ be a closed, proper, non-trivial subspace. Let $P$ be the orthogonal projection to $H_{1}$.

  (i) Prove that $P$ is self-adjoint.

  (ii) Determine the spectrum $\sigma(P)$ and the point spectrum $\sigma_{p}(P)$ of $P$.

  (iii) Find a necessary and sufficient condition on $H_{1}$ for $P$ to be compact.

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• Paper 2, Section II, F

  Logic and Set Theory

  Part II, 2016

  Define the von Neumann hierarchy of sets $V_{\alpha}$, and show that each $V_{\alpha}$ is a transitive set. Explain what is meant by saying that a binary relation on a set is well-founded and extensional. State Mostowski's Theorem.

  Let $r$ be the binary relation on $\omega$ defined by: $\langle m, n\rangle \in r$ if and only if $2^{m}$ appears in the base-2 expansion of $n$ (i.e., the unique expression for $n$ as a sum of distinct powers of 2 ). Show that $r$ is well-founded and extensional. To which transitive set is $(\omega, r)$ isomorphic? Justify your answer.

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• Paper 3, Section II, F

  Logic and Set Theory

  Part II, 2016

  State the Completeness Theorem for the first-order predicate calculus, and deduce the Compactness Theorem.

  Let $\mathbb{T}$ be a first-order theory over a signature $\Sigma$ whose axioms all have the form $(\forall \vec{x}) \phi$ where $\vec{x}$ is a (possibly empty) string of variables and $\phi$ is quantifier-free. Show that every substructure of a $\mathbb{T}$-model is a $\mathbb{T}$-model, and deduce that if $\mathbb{T}$ is consistent then it has a model in which every element is the interpretation of a closed term of $\mathcal{L}(\Sigma)$. $[$ You may assume the result that if $B$ is a substructure of $A$ and $\phi$ is a quantifier-free formula with $n$ free variables, then $\llbracket \phi \rrbracket_{B}=\llbracket \phi \rrbracket_{A} \cap B^{n}$.]

  Now suppose $\mathbb{T} \vdash(\exists x) \psi$ where $\psi$ is a quantifier-free formula with one free variable $x$. Show that there is a finite list $\left(t_{1}, t_{2}, \ldots, t_{n}\right)$ of closed terms of $\mathcal{L}(\Sigma)$ such that

  $$\mathbb{T} \vdash\left(\psi\left[t_{1} / x\right] \vee \psi\left[t_{2} / x\right] \vee \cdots \vee \psi\left[t_{n} / x\right]\right)$$

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• Paper 1, Section II, F

  Logic and Set Theory

  Part II, 2016

  Which of the following statements are true? Justify your answers. (a) Every ordinal is of the form $\alpha+n$, where $\alpha$ is a limit ordinal and $n \in \omega$. (b) Every ordinal is of the form $\omega^{\alpha} \cdot m+n$, where $\alpha$ is an ordinal and $m, n \in \omega$. (c) If $\alpha=\omega \cdot \alpha$, then $\alpha=\omega^{\omega} \cdot \beta$ for some $\beta$. (d) If $\alpha=\omega^{\alpha}$, then $\alpha$ is uncountable. (e) If $\alpha>1$ and $\alpha=\alpha^{\omega}$, then $\alpha$ is uncountable.

  [Standard laws of ordinal arithmetic may be assumed, but if you use the Division Algorithm you should prove it.]

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• Paper 4, Section II, F

  Logic and Set Theory

  Part II, 2016

  (a) State Zorn's Lemma, and use it to prove that every nontrivial distributive lattice $L$ admits a lattice homomorphism $L \rightarrow\{0,1\}$.

  (b) Let $S$ be a propositional theory in a given language $\mathcal{L}$. Sketch the way in which the equivalence classes of formulae of $\mathcal{L}$, modulo $S$-provable equivalence, may be made into a Boolean algebra. [Detailed proofs are not required, but you should define the equivalence relation explicitly.]

  (c) Hence show how the Completeness Theorem for propositional logic may be deduced from the result of part (a).

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• Paper 4, Section I, B

  Mathematical Biology

  Part II, 2016

  A stochastic birth-death process is given by the master equation

  $$\frac{d p_{n}}{d t}=\lambda\left(p_{n-1}-p_{n}\right)+\mu\left[(n-1) p_{n-1}-n p_{n}\right]+\beta\left[(n+1) p_{n+1}-n p_{n}\right]$$

  where $p_{n}(t)$ is the probability that there are $n$ individuals in the population at time $t$ for $n=0,1,2, \ldots$ and $p_{n}=0$ for $n<0$. Give a brief interpretation of $\lambda, \mu$ and $\beta$.

  Derive an equation for $\frac{\partial \phi}{\partial t}$, where $\phi$ is the generating function

  $$\phi(s, t)=\sum_{n=0}^{\infty} s^{n} p_{n}(t)$$

  Now assume that $\beta>\mu$. Show that at steady state

  $$\phi=\left(\frac{\beta-\mu}{\beta-\mu s}\right)^{\lambda / \mu}$$

  and find the corresponding mean and variance.

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• Paper 3, Section $\mathbf{I}$, B

  Mathematical Biology

  Part II, 2016

  A delay model for a population of size $N_{t}$ at discrete time $t$ is given by

  $$N_{t+1}=\max \left\{\left(r-N_{t-1}^{2}\right) N_{t}, 0\right\}$$

  Show that for $r>1$ there is a non-trivial equilibrium, and analyse its stability. Show that, as $r$ is increased from 1 , the equilibrium loses stability at $r=3 / 2$ and find the approximate periodicity close to equilibrium at this point.

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• Paper 2, Section I, B

  Mathematical Biology

  Part II, 2016

  (a) The populations of two competing species satisfy

  $$\begin{aligned} \frac{d N_{1}}{d t} &=N_{1}\left[b_{1}-\lambda\left(N_{1}+N_{2}\right)\right] \\ \frac{d N_{2}}{d t} &=N_{2}\left[b_{2}-\lambda\left(N_{1}+N_{2}\right)\right] \end{aligned}$$

  where $b_{1}>b_{2}>0$ and $\lambda>0$. Sketch the phase diagram (limiting attention to $\left.N_{1}, N_{2} \geqslant 0\right)$.

  The relative abundance of species 1 is defined by $U=N_{1} /\left(N_{1}+N_{2}\right)$. Show that

  $$\frac{d U}{d t}=A U(1-U)$$

  where $A$ is a constant that should be determined.

  (b) Consider the spatial system

  $$\frac{\partial u}{\partial t}=u(1-u)+D \frac{\partial^{2} u}{\partial x^{2}}$$

  and consider a travelling-wave solution of the form $u(x, t)=f(x-c t)$ representing one species $(u=1)$ invading territory previously occupied by another species $(u=0)$. By linearising near the front of the invasion, show that the wave speed is given by $c=2 \sqrt{D}$.

  [You may assume that the solution to the full nonlinear system will settle to the slowest possible linear wave speed.]

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• Paper 1, Section I, B

  Mathematical Biology

  Part II, 2016

  Consider an epidemic model where susceptibles are vaccinated at per capita rate $v$, but immunity (from infection or vaccination) is lost at per capita rate $b$. The system is given by

  $$\begin{aligned} &\frac{d S}{d t}=-r I S+b(N-I-S)-v S \\ &\frac{d I}{d t}=r I S-a I \end{aligned}$$

  where $S(t)$ are the susceptibles, $I(t)$ are the infecteds, $N$ is the total population size and all parameters are positive. The basic reproduction ratio $R_{0}=r N / a$ satisfies $R_{0}>1$.

  Find the critical vaccination rate $v_{c}$, in terms of $b$ and $R_{0}$, such that the system has an equilibrium with the disease present if $v<v_{c}$. Show that this equilibrium is stable when it exists.

  Find the long-term outcome for $S$ and $I$ if $v>v_{c}$.

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• Paper 3, Section II, B

  Mathematical Biology

  Part II, 2016

  The Fitzhugh-Nagumo model is given by

  $$\begin{aligned} \dot{u} &=c\left(v+u-\frac{1}{3} u^{3}+z(t)\right) \\ \dot{v} &=-\frac{1}{c}(u-a+b v) \end{aligned}$$

  where $\left(1-\frac{2}{3} b\right)<a<1,0<b \leqslant 1$ and $c \gg 1$.

  For $z(t)=0$, by considering the nullclines in the $(u, v)$-plane, show that there is a unique equilibrium. Sketch the phase diagram

  At $t=0$ the system is at the equilibrium, and $z(t)$ is then 'switched on' to be $z(t)=-V_{0}$ for $t>0$, where $V_{0}$ is a constant. Describe carefully how suitable choices of $V_{0}>0$ can represent a system analogous to (i) a neuron firing once, and (ii) a neuron firing repeatedly. Illustrate your answer with phase diagrams and also plots of $v$ against $t$ for each case. Find the threshold for $V_{0}$ that separates these cases. Comment briefly from a biological perspective on the behaviour of the system when $a=1-\frac{2}{3} b+\epsilon b$ and $0<\epsilon \ll 1$.

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• Paper 4, Section II, B

  Mathematical Biology

  Part II, 2016

  The population densities of two types of cell are given by $U(x, t)$ and $V(x, t)$. The system is described by the equations

  $$\begin{aligned} \frac{\partial U}{\partial t} &=\alpha U(1-U)+\chi \frac{\partial}{\partial x}\left(U \frac{\partial V}{\partial x}\right)+D \frac{\partial^{2} U}{\partial x^{2}} \\ \frac{\partial V}{\partial t} &=V(1-V)-\beta U V+\frac{\partial^{2} V}{\partial x^{2}} \end{aligned}$$

  where $\alpha, \beta, \chi$ and $D$ are positive constants.

  (a) Identify the terms which involve interaction between the cell types, and briefly describe what each of these terms might represent.

  (b) Consider the system without spatial dynamics. Find the condition on $\beta$ for there to be a non-trivial spatially homogeneous solution that is stable to spatially invariant disturbances.

  (c) Consider now the full spatial system, and consider small spatial perturbations proportional to $\cos (k x)$ of the solution found in part (b). Show that for sufficiently large $\chi$ (the precise threshold should be found) the spatially homogeneous solution is stable to perturbations with either small or large wavenumber, but is unstable to perturbations at some intermediate wavenumber.

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• Paper 2, Section II, F

  Number Fields

  Part II, 2016

  (a) Prove that $5+2 \sqrt{6}$ is a fundamental unit in $\mathbb{Q}(\sqrt{6})$. [You may not assume the continued fraction algorithm.]

  (b) Determine the ideal class group of $\mathbb{Q}(\sqrt{-55})$.

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• Paper 1, Section II, F

  Number Fields

  Part II, 2016

  (a) Let $f(X) \in \mathbb{Q}[X]$ be an irreducible polynomial of degree $n, \theta \in \mathbb{C}$ a root of $f$, and $K=\mathbb{Q}(\theta)$. Show that $\operatorname{disc}(f)=\pm N_{K / \mathbb{Q}}\left(f^{\prime}(\theta)\right)$.

  (b) Now suppose $f(X)=X^{n}+a X+b$. Write down the matrix representing multiplication by $f^{\prime}(\theta)$ with respect to the basis $1, \theta, \ldots, \theta^{n-1}$ for $K$. Hence show that

  $$\operatorname{disc}(f)=\pm\left((1-n)^{n-1} a^{n}+n^{n} b^{n-1}\right)$$

  (c) Suppose $f(X)=X^{4}+X+1$. Determine $\mathcal{O}_{K}$. [You may quote any standard result, as long as you state it clearly.]

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• Paper 4, Section II, F

  Number Fields

  Part II, 2016

  Let $K$ be a number field, and $p$ a prime in $\mathbb{Z}$. Explain what it means for $p$ to be inert, to split completely, and to be ramified in $K$.

  (a) Show that if $[K: \mathbb{Q}]>2$ and $\mathcal{O}_{K}=\mathbb{Z}[\alpha]$ for some $\alpha \in K$, then 2 does not split completely in $K$.

  (b) Let $K=\mathbb{Q}(\sqrt{d})$, with $d \neq 0,1$ and $d$ square-free. Determine, in terms of $d$, whether $p=2$ splits completely, is inert, or ramifies in $K$. Hence show that the primes which ramify in $K$ are exactly those which divide $D_{K}$.

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• Paper 3, Section I, I

  Number Theory

  Part II, 2016

  Show that the exact power of a prime $p$ dividing $N !$ is $\sum_{j=1}^{\infty}\left\lfloor\frac{N}{p^{j}}\right\rfloor$. By considering the prime factorisation of $\left(\begin{array}{c}2 n \\ n\end{array}\right)$, show that

  $$\frac{4^{n}}{2 n+1} \leqslant\left(\begin{array}{c} 2 n \\ n \end{array}\right) \leqslant(2 n)^{\pi(2 n)}$$

  Setting $n=\left\lfloor\frac{x}{2}\right\rfloor$, deduce that for $x$ sufficiently large

  $$\pi(x)>\frac{\left\lfloor\frac{x}{2}\right\rfloor \log 3}{\log x}>\frac{x}{2 \log x}$$

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• Paper 4 , Section I, I

  Number Theory

  Part II, 2016

  Compute the continued fraction expansion of $\sqrt{14}$, and use it to find two solutions to $x^{2}-14 y^{2}=2$ where $x$ and $y$ are positive integers.

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• Paper 2, Section I, I

  Number Theory

  Part II, 2016

  Define the Legendre symbol and the Jacobi symbol. Compute the Jacobi symbols $\left(\frac{202}{11189}\right)$ and $\left(\frac{974}{1001}\right)$, stating clearly any properties of these symbols that you use.

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• Paper 4, Section II, 10I

  Number Theory

  Part II, 2016

  (a) Define Euler's totient function $\phi(n)$ and show that $\sum_{d \mid n} \phi(d)=n$.

  (b) State Lagrange's theorem concerning roots of polynomials mod $p$.

  (c) Let $p$ be a prime. Proving any results you need about primitive roots, show that $x^{m} \equiv 1(\bmod p)$ has exactly $(m, p-1)$ roots.

  (d) Show that if $p$ and $3 p-2$ are both primes then $N=p(3 p-2)$ is a Fermat pseudoprime for precisely a third of all bases.

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• Paper 1, Section I, I

  Number Theory

  Part II, 2016

  Define the Riemann zeta function $\zeta(s)$ for $\operatorname{Re}(s)>1$. State and prove the alternative formula for $\zeta(s)$ as an Euler product. Hence or otherwise show that $\zeta(s) \neq 0$ for $\operatorname{Re}(s)>1$.

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• Paper 3, Section II, I

  Number Theory

  Part II, 2016

  What does it mean for a positive definite binary quadratic form to be reduced?

  Prove that every positive definite binary quadratic form is equivalent to a reduced form, and that there are only finitely many reduced forms with given discriminant.

  State a criterion for a positive integer $n$ to be represented by a positive definite binary quadratic form with discriminant $d<0$, and hence determine which primes $p$ are represented by $x^{2}+x y+7 y^{2}$.

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• Paper 4, Section II, B

  Numerical Analysis

  Part II, 2016

  (a) Describe an implementation of the power method for determining the eigenvalue of largest modulus and its associated eigenvector for a matrix that has a unique eigenvalue of largest modulus.

  Now let $A$ be a real $n \times n$ matrix with distinct eigenvalues satisfying $\left|\lambda_{n}\right|=\left|\lambda_{n-1}\right|$ and $\left|\lambda_{n}\right|>\left|\lambda_{i}\right|, i=1, \ldots, n-2$. The power method is applied to $A$, with an initial condition $\mathbf{x}^{(0)}=\sum_{i=1}^{n} c_{i} \mathbf{w}_{i}$ such that $c_{n-1} c_{n} \neq 0$, where $\mathbf{w}_{i}$ is the eigenvector associated with $\lambda_{i}$. Show that the power method does not converge. Explain why $\mathbf{x}^{(k)}, \mathbf{x}^{(k+1)}$ and $\mathbf{x}^{(k+2)}$ become linearly dependent as $k \rightarrow \infty$.

  (b) Consider the following variant of the power method, called the two-stage power method, applied to the matrix $A$ of part (a):

  1. Pick $\mathbf{x}^{(0)} \in \mathbb{R}^{n}$ satisfying $\left\|\mathbf{x}^{(0)}\right\|=1$. Let $0<\varepsilon \ll 1$. Set $k=0$ and $\mathbf{x}^{(1)}=A \mathbf{x}^{(0)}$.

  2. Calculate $\mathbf{x}^{(k+2)}=A \mathbf{x}^{(k+1)}$ and calculate $\alpha, \beta$ that minimise

  $$f(\alpha, \beta)=\left\|\mathbf{x}^{(k+2)}+\alpha \mathbf{x}^{(k+1)}+\beta \mathbf{x}^{(k)}\right\| .$$

  2. If $f(\alpha, \beta) \leqslant \varepsilon$, solve $\lambda^{2}+\alpha \lambda+\beta=0$ and let the roots be $\lambda_{1}$ and $\lambda_{2}$. They are accepted as eigenvalues of $A$, and the corresponding eigenvectors are estimated as $\mathbf{x}^{(k+1)}-\lambda_{2} \mathbf{x}^{(k)}$ and $\mathbf{x}^{(k+1)}-\lambda_{1} \mathbf{x}^{(k)}$.

  3. Otherwise, divide $\mathbf{x}^{(k+2)}$ and $\mathbf{x}^{(k+1)}$ by the current value of $\left\|\mathbf{x}^{(k+1)}\right\|$, increase $k$ by 1 and return to Step $\mathbf{1}$.

  Explain the justification behind Step 2 of the algorithm.

  (c) Let $n=3$, and suppose that, for a large value of $k$, the two-stage power method yields the vectors

  $$\mathbf{y}_{k}=\mathbf{x}^{(k)}=\left(\begin{array}{l} 1 \\ 1 \\ 1 \end{array}\right), \quad \mathbf{y}_{k+1}=A \mathbf{x}^{(k)}=\left(\begin{array}{c} 2 \\ 3 \\ 4 \end{array}\right), \quad \mathbf{y}_{k+2}=A^{2} \mathbf{x}^{(k)}=\left(\begin{array}{c} 2 \\ 4 \\ 6 \end{array}\right)$$

  Find two eigenvalues of $A$ and the corresponding eigenvectors.

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• Paper 2, Section II, B

  Numerical Analysis

  Part II, 2016

  (a) The advection equation

  $$u_{t}=u_{x}, \quad 0 \leqslant x \leqslant 1, t \geqslant 0$$

  is discretised using an equidistant grid with stepsizes $\Delta x=h$ and $\Delta t=k$. The spatial derivatives are approximated with central differences and the resulting ODEs are approximated with the trapezoidal rule. Write down the relevant difference equation for determining $\left(u_{m}^{n+1}\right)$ from $\left(u_{m}^{n}\right)$. What is the name of this scheme? What is the local truncation error?

  The boundary condition is periodic, $u(0, t)=u(1, t)$. Explain briefly how to write the discretised scheme in the form $B \mathbf{u}^{n+1}=C \mathbf{u}^{n}$, where the matrices $B$ and $C$, to be identified, have a circulant form. Using matrix analysis, find the range of $\mu=\Delta t / \Delta x$ for which the scheme is stable. [Standard results may be used without proof if quoted carefully.]

  [Hint: An $n \times n$ circulant matrix has the form

  $$A=\left(\begin{array}{cccc} a_{0} & a_{1} & \cdots & a_{n-1} \\ a_{n-1} & \ddots & \ddots & \vdots \\ \vdots & \ddots & \ddots & a_{1} \\ a_{1} & \cdots & a_{n-1} & a_{0} \end{array}\right)$$

  All such matrices have the same set of eigenvectors $\mathbf{v}_{\ell}=\left(\omega^{j \ell}\right)_{j=0}^{n-1}, \ell=0,1, \ldots, n-1$, where $\omega=e^{2 \pi i / n}$, and the corresponding eigenvalues are $\lambda_{\ell}=\sum_{k=0}^{n-1} a_{k} \omega^{k \ell}$.]

  (b) Consider the advection equation on the unit square

  $$u_{t}=a u_{x}+b u_{y}, \quad 0 \leqslant x, y \leqslant 1, t \geqslant 0$$

  where $u$ satisfies doubly periodic boundary conditions, $u(0, y)=u(1, y), u(x, 0)=u(x, 1)$, and $a(x, y)$ and $b(x, y)$ are given doubly periodic functions. The system is discretised with the Crank-Nicolson scheme, with central differences for the space derivatives, using an equidistant grid with stepsizes $\Delta x=\Delta y=h$ and $\Delta t=k$. Write down the relevant difference equation, and show how to write the scheme in the form

  $$\mathbf{u}^{n+1}=\left(I-\frac{1}{4} \mu A\right)^{-1}\left(I+\frac{1}{4} \mu A\right) \mathbf{u}^{n}$$

  where the matrix $A$ should be identified. Describe how (*) can be approximated by Strang splitting, and explain the advantages of doing so.

  [Hint: Inversion of the matrix $B$ in part (a) has a similar computational cost to that of a tridiagonal matrix.]

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• Paper 1, Section II, B

  Numerical Analysis

  Part II, 2016

  (a) Consider the periodic function

  $$f(x)=5+2 \cos \left(2 \pi x-\frac{\pi}{2}\right)+3 \cos (4 \pi x)$$

  on the interval $[0,1]$. The $N$-point discrete Fourier transform of $f$ is defined by

  $$F_{N}(n)=\frac{1}{N} \sum_{k=0}^{N-1} f_{k} \omega_{N}^{-n k}, \quad n=0,1, \ldots, N-1$$

  where $\omega_{N}=e^{2 \pi i / N}$ and $f_{k}=f(k / N)$. Compute $F_{4}(n), n=0, \ldots, 3$, using the Fast Fourier Transform (FFT). Compare your result with what you get by computing $F_{4}(n)$ directly from $(*)$. Carefully explain all your computations.

  (b) Now let $f$ be any analytic function on $\mathbb{R}$ that is periodic with period 1 . The continuous Fourier transform of $f$ is defined by

  $$\hat{f}_{n}=\int_{0}^{1} f(\tau) e^{-2 \pi i n \tau} d \tau, \quad n \in \mathbb{Z}$$

  Use integration by parts to show that the Fourier coefficients $\hat{f}_{n}$ decay spectrally.

  Explain what it means for the discrete Fourier transform of $f$ to approximate the continuous Fourier transform with spectral accuracy. Prove that it does so.

  What can you say about the behaviour of $F_{N}(N-n)$ as $N \rightarrow \infty$ for fixed $n$ ?

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• Paper 3, Section II, B

  Numerical Analysis

  Part II, 2016

  (a) Define the Jacobi and Gauss-Seidel iteration schemes for solving a linear system of the form $A \mathbf{u}=\mathbf{b}$, where $\mathbf{u}, \mathbf{b} \in \mathbb{R}^{M}$ and $A \in \mathbb{R}^{M \times M}$, giving formulae for the corresponding iteration matrices $H_{J}$ and $H_{G S}$ in terms of the usual decomposition $A=L_{0}+D+U_{0}$.

  Show that both iteration schemes converge when $A$ results from discretization of Poisson's equation on a square with the five-point formula, that is when

  $$A=\left[\begin{array}{rrrrr} S & I & & & \\ I & S & I & & \\ & \ddots & \ddots & \ddots & \\ & & I & S & I \\ & & & I & S \end{array}\right], \quad S=\left[\begin{array}{rrrrr} -4 & 1 & & & \\ 1 & -4 & 1 & & \\ & \ddots & \ddots & \ddots & \\ & & 1 & -4 & 1 \\ & & & 1 & -4 \end{array}\right] \in \mathbb{R}^{m \times m}$$

  and $M=m^{2}$. [You may state the Householder-John theorem without proof.]

  (b) For the matrix $A$ given in $(*)$ :

  (i) Calculate the eigenvalues of $H_{J}$ and deduce its spectral radius $\rho\left(H_{J}\right)$.

  (ii) Show that each eigenvector $\mathbf{q}$ of $H_{G S}$ is related to an eigenvector $\mathbf{p}$ of $H_{J}$ by a transformation of the form $q_{i, j}=\alpha^{i+j} p_{i, j}$ for a suitable value of $\alpha$.

  (iii) Deduce that $\rho\left(H_{G S}\right)=\rho^{2}\left(H_{J}\right)$. What is the significance of this result for the two iteration schemes?

  [Hint: You may assume that the eigenvalues of the matrix $A$ in $(*)$ are

  $$\lambda_{k, \ell}=-4\left(\sin ^{2} \frac{x}{2}+\sin ^{2} \frac{y}{2}\right), \quad \text { where } x=\frac{k \pi h}{m+1}, y=\frac{\ell \pi h}{m+1}, \quad k, \ell=1, \ldots, m,$$

  with corresponding eigenvectors $\left.\mathbf{v}=\left(v_{i, j}\right), \quad v_{i, j}=\sin i x \sin j y, \quad i, j=1, \ldots, m . \quad\right]$

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• Paper 3, Section II, K

  Optimization and Control

  Part II, 2016

  Consider the system in scalar variables, for $t=1,2, \ldots, h$ :

  $$\begin{aligned} x_{t} &=x_{t-1}+u_{t-1} \\ y_{t} &=x_{t-1}+\eta_{t} \\ \hat{x}_{0} &=x_{0}+\eta_{0} \end{aligned}$$

  where $\hat{x}_{0}$ is given, $y_{t}, u_{t}$ are observed at $t$, but $x_{0}, x_{1}, \ldots$ and $\eta_{0}, \eta_{1}, \ldots$ are unobservable, and $\eta_{0}, \eta_{1}, \ldots$ are independent random variables with mean 0 and variance $v$. Define $\hat{x}_{t-1}$ to be the estimator of $x_{t-1}$ with minimum variance amongst all estimators that are unbiased and linear functions of $W_{t-1}=\left(\hat{x}_{0}, y_{1}, \ldots, y_{t-1}, u_{0}, \ldots, u_{t-2}\right)$. Suppose $\hat{x}_{t-1}=a^{T} W_{t-1}$ and its variance is $V_{t-1}$. After observation at $t$ of $\left(y_{t}, u_{t-1}\right)$, a new unbiased estimator of $x_{t-1}$, linear in $W_{t}$, is expressed

  $$x_{t-1}^{*}=(1-H) b^{T} W_{t-1}+H y_{t}$$

  Find $b$ and $H$ to minimize the variance of $x_{t-1}^{*}$. Hence find $\hat{x}_{t}$ in terms of $\hat{x}_{t-1}, y_{t}, u_{t-1}$, $V_{t-1}$ and $v$. Calculate $V_{h}$.

  Suppose $\eta_{0}, \eta_{1}, \ldots$ are Gaussian and thus $\hat{x}_{t}=E\left[x_{t} \mid W_{t}\right]$. Consider minimizing $E\left[x_{h}^{2}+\sum_{t=0}^{h-1} u_{t}^{2}\right]$, under the constraint that the control $u_{t}$ can only depend on $W_{t}$. Show that the value function of dynamic programming for this problem can be expressed

  $$F\left(W_{t}\right)=\Pi_{t} \hat{x}_{t}^{2}+\cdots$$

  where $F\left(W_{h}\right)=\hat{x}_{h}^{2}+V_{h}$ and $+\cdots$ is independent of $W_{t}$ and linear in $v$.

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• Paper 4, Section II, K

  Optimization and Control

  Part II, 2016

  State transversality conditions that can be used with Pontryagin's maximum principle and say when they are helpful.

  Given $T$, it is desired to maximize $c_{1} x_{1}(T)+c_{2} x_{2}(T)$, where

  $$\begin{aligned} &\dot{x}_{1}=u_{1}\left(a_{1} x_{1}+a_{2} x_{2}\right), \\ &\dot{x}_{2}=u_{2}\left(a_{1} x_{1}+a_{2} x_{2}\right), \end{aligned}$$

  and $u=\left(u_{1}, u_{2}\right)$ is a time-varying control such that $u_{1} \geqslant 0, u_{2} \geqslant 0$ and $u_{1}+u_{2}=1$. Suppose that $x_{1}(0)$ and $x_{2}(0)$ are positive, and that $0<a_{2}<a_{1}$ and $0<c_{1}<c_{2}$. Find the optimal control at times close to $T$. Show that over $[0, T]$ the optimal control is constant, or makes exactly one switch, the latter happening if and only if

  $$c_{2} e^{a_{2} T}<c_{1}+\frac{a_{1} c_{2}}{a_{2}}\left(e^{a_{2} T}-1\right)$$

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• Paper 2, Section II, K

  Optimization and Control

  Part II, 2016

  Consider a Markov decision problem with finite state space $X$, value function $F$ and dynamic programming equation $F=\mathcal{L} F$, where

  $$(\mathcal{L} \phi)(i)=\min _{a \in\{0,1\}}\left\{c(i, a)+\beta \sum_{j \in X} P_{i j}(a) \phi(j)\right\} .$$

  Suppose $0<\beta<1$, and $|c(i, a)| \leqslant B$ for all $i \in X, a \in\{0,1\}$. Prove there exists a deterministic stationary Markov policy that is optimal, explaining what the italicised words mean.

  Let $F_{n}=\mathcal{L}^{n} F_{0}$, where $F_{0}=0$, and $M_{n}=\max _{i \in X}\left|F(i)-F_{n}(i)\right|$. Prove that

  $$M_{n} \leqslant \beta M_{n-1} \leqslant \beta^{n} B /(1-\beta) .$$

  Deduce that the value iteration algorithm converges to an optimal policy in a finite number of iterations.

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• Paper 1, Section II, A

  Principles of Quantum Mechanics

  Part II, 2016

  A particle in one dimension has position and momentum operators $\hat{x}$ and $\hat{p}$ whose eigenstates obey

  $$\left\langle x \mid x^{\prime}\right\rangle=\delta\left(x-x^{\prime}\right), \quad\left\langle p \mid p^{\prime}\right\rangle=\delta\left(p-p^{\prime}\right), \quad\langle x \mid p\rangle=(2 \pi \hbar)^{-1 / 2} e^{i x p / \hbar}$$

  For a state $|\psi\rangle$, define the position-space and momentum-space wavefunctions $\psi(x)$ and $\tilde{\psi}(p)$ and show how each of these can be expressed in terms of the other.

  Write down the translation operator $U(\alpha)$ and check that your expression is consistent with the property $U(\alpha)|x\rangle=|x+\alpha\rangle$. For a state $|\psi\rangle$, relate the position-space and momentum-space wavefunctions for $U(\alpha)|\psi\rangle$ to $\psi(x)$ and $\tilde{\psi}(p)$ respectively.

  Now consider a harmonic oscillator with mass $m$, frequency $\omega$, and annihilation and creation operators

  $$a=\left(\frac{m \omega}{2 \hbar}\right)^{1 / 2}\left(\hat{x}+\frac{i}{m \omega} \hat{p}\right), \quad a^{\dagger}=\left(\frac{m \omega}{2 \hbar}\right)^{1 / 2}\left(\hat{x}-\frac{i}{m \omega} \hat{p}\right)$$

  Let $\psi_{n}(x)$ and $\tilde{\psi}_{n}(p)$ be the wavefunctions corresponding to the normalised energy eigenstates $|n\rangle$, where $n=0,1,2, \ldots$.

  (i) Express $\psi_{0}(x-\alpha)$ explicitly in terms of the wavefunctions $\psi_{n}(x)$.

  (ii) Given that $\tilde{\psi}_{n}(p)=f_{n}(u) \tilde{\psi}_{0}(p)$, where the $f_{n}$ are polynomials and $u=(2 / \hbar m \omega)^{1 / 2} p$, show that

  $$e^{-i \gamma u}=e^{-\gamma^{2} / 2} \sum_{n=0}^{\infty} \frac{\gamma^{n}}{\sqrt{n !}} f_{n}(u) \text { for any real } \gamma$$

  [You may quote standard results for a harmonic oscillator. You may also use, without proof, $e^{A+B}=e^{A} e^{B} e^{-\frac{1}{2}[A, B]}$ for operators $A$ and $B$ which each commute with $\left.[A, B] .\right]$

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• Paper 3, Section II, 31 A

  Principles of Quantum Mechanics

  Part II, 2016

  A three-dimensional oscillator has Hamiltonian

  $$H=\frac{1}{2 m}\left(\hat{p}_{1}^{2}+\hat{p}_{2}^{2}+\hat{p}_{3}^{2}\right)+\frac{1}{2} m \omega^{2}\left(\alpha^{2} \hat{x}_{1}^{2}+\beta^{2} \hat{x}_{2}^{2}+\gamma^{2} \hat{x}_{3}^{2}\right),$$

  where the constants $m, \omega, \alpha, \beta, \gamma$ are real and positive. Assuming a unique ground state, construct the general normalised eigenstate of $H$ and give a formula for its energy eigenvalue. [You may quote without proof results for a one-dimensional harmonic oscillator of mass $m$ and frequency $\omega$ that follow from writing $\hat{x}=(\hbar / 2 m \omega)^{1 / 2}\left(a+a^{\dagger}\right)$ and $\left.\hat{p}=(\hbar m \omega / 2)^{1 / 2} i\left(a^{\dagger}-a\right) .\right]$

  List all states in the four lowest energy levels of $H$ in the cases:

  (i) $\alpha<\beta<\gamma<2 \alpha$;

  (ii) $\alpha=\beta$ and $\gamma=\alpha+\epsilon$, where $0<\epsilon \ll \alpha$.

  Now consider $H$ with $\alpha=\beta=\gamma=1$ subject to a perturbation

  $$\lambda m \omega^{2}\left(\hat{x}_{1} \hat{x}_{2}+\hat{x}_{2} \hat{x}_{3}+\hat{x}_{3} \hat{x}_{1}\right),$$

  where $\lambda$ is small. Compute the changes in energies for the ground state and the states at the first excited level of the original Hamiltonian, working to the leading order at which nonzero corrections occur. [You may quote without proof results from perturbation theory.]

  Explain briefly why some energy levels of the perturbed Hamiltonian will be exactly degenerate. [Hint: Compare with (ii) above.]

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• Paper 4, Section II, A

  Principles of Quantum Mechanics

  Part II, 2016

  (a) Consider a quantum system with Hamiltonian $H=H_{0}+V$, where $H_{0}$ is independent of time. Define the interaction picture corresponding to this Hamiltonian and derive an expression for the time derivative of an operator in the interaction picture, assuming it is independent of time in the Schrödinger picture.

  (b) The Pauli matrices $\boldsymbol{\sigma}=\left(\sigma_{1}, \sigma_{2}, \sigma_{3}\right)$ satisfy

  $$\sigma_{i} \sigma_{j}=\delta_{i j}+i \epsilon_{i j k} \sigma_{k}$$

  Explain briefly how these properties allow $\sigma$ to be used to describe a quantum system with spin $\frac{1}{2}$.

  (c) A particle with spin $\frac{1}{2}$ has position and momentum operators $\hat{\mathbf{x}}=\left(\hat{x}_{1}, \hat{x}_{2}, \hat{x}_{3}\right)$ and $\hat{\mathbf{p}}=\left(\hat{p}_{1}, \hat{p}_{2}, \hat{p}_{3}\right)$. The unitary operator corresponding to a rotation through an angle $\theta$ about an axis $\mathbf{n}$ is $U=\exp (-i \theta \mathbf{n} \cdot \mathbf{J} / \hbar)$ where $\mathbf{J}$ is the total angular momentum. Check this statement by considering the effect of an infinitesimal rotation on $\hat{\mathbf{x}}, \hat{\mathbf{p}}$ and $\boldsymbol{\sigma}$.

  (d) Suppose that the particle in part (c) has Hamiltonian $H=H_{0}+V$ with

  $$H_{0}=\frac{1}{2 m} \hat{\mathbf{p}}^{2}+\alpha \mathbf{L} \cdot \boldsymbol{\sigma} \quad \text { and } \quad V=B \sigma_{3}$$

  where $\mathbf{L}$ is the orbital angular momentum and $\alpha, B$ are constants. Show that all components of $\mathbf{J}$ are independent of time in the interaction picture. Is this true in the Heisenberg picture?

  [You may quote commutation relations of $\mathbf{L}$ with $\hat{\mathbf{x}}$ and $\hat{\mathbf{p}}$.]

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• Paper 2, Section II, A

  Principles of Quantum Mechanics

  Part II, 2016

  (a) Let $|j m\rangle$ be standard, normalised angular momentum eigenstates with labels specifying eigenvalues for $\mathbf{J}^{2}$ and $J_{3}$. Taking units in which $\hbar=1$,

  $$J_{\pm}|j m\rangle=\{(j \mp m)(j \pm m+1)\}^{1 / 2}|j m \pm 1\rangle .$$

  Check the coefficients above by computing norms of states, quoting any angular momentum commutation relations that you require.

  (b) Two particles, each of spin $s>0$, have combined spin states $|J M\rangle$. Find expressions for all such states with $M=2 s-1$ in terms of product states.

  (c) Suppose that the particles in part (b) move about their centre of mass with a spatial wavefunction that is a spherically symmetric function of relative position. If the particles are identical, what spin states $|J 2 s-1\rangle$ are allowed? Justify your answer.

  (d) Now consider two particles of spin 1 that are not identical and are both at rest. If the 3-component of the spin of each particle is zero, what is the probability that their total, combined spin is zero?

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• Paper 3, Section II, J

  Principles of Statistics

  Part II, 2016

  Let $X_{1}, \ldots, X_{n}$ be i.i.d. random variables from a $N(\theta, 1)$ distribution, $\theta \in \mathbb{R}$, and consider a Bayesian model $\theta \sim N\left(0, v^{2}\right)$ for the unknown parameter, where $v>0$ is a fixed constant.

  (a) Derive the posterior distribution $\Pi\left(\cdot \mid X_{1}, \ldots, X_{n}\right)$ of $\theta \mid X_{1}, \ldots, X_{n}$.

  (b) Construct a credible set $C_{n} \subset \mathbb{R}$ such that

  (i) $\Pi\left(C_{n} \mid X_{1}, \ldots, X_{n}\right)=0.95$ for every $n \in \mathbb{N}$, and

  (ii) for any $\theta_{0} \in \mathbb{R}$,

  $$P_{\theta_{0}}^{\mathbb{N}}\left(\theta_{0} \in C_{n}\right) \rightarrow 0.95 \quad \text { as } n \rightarrow \infty,$$

  where $P_{\theta}^{\mathbb{N}}$ denotes the distribution of the infinite sequence $X_{1}, X_{2}, \ldots$ when drawn independently from a fixed $N(\theta, 1)$ distribution.

  [You may use the central limit theorem.]

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• Paper 2, Section II,

  Principles of Statistics

  Part II, 2016

  (a) State and prove the Cramér-Rao inequality in a parametric model $\{f(\theta): \theta \in \Theta\}$, where $\Theta \subseteq \mathbb{R}$. [Necessary regularity conditions on the model need not be specified.]

  (b) Let $X_{1}, \ldots, X_{n}$ be i.i.d. Poisson random variables with unknown parameter $E X_{1}=\theta>0$. For $\bar{X}_{n}=(1 / n) \sum_{i=1}^{n} X_{i}$ and $S^{2}=(n-1)^{-1} \sum_{i=1}^{n}\left(X_{i}-\bar{X}_{n}\right)^{2}$ define

  $$T_{\alpha}=\alpha \bar{X}_{n}+(1-\alpha) S^{2}, \quad 0 \leqslant \alpha \leqslant 1$$

  Show that $\operatorname{Var}_{\theta}\left(T_{\alpha}\right) \geqslant \operatorname{Var}_{\theta}\left(\bar{X}_{n}\right)$ for all values of $\alpha, \theta$.

  Now suppose $\tilde{\theta}=\tilde{\theta}\left(X_{1}, \ldots, X_{n}\right)$ is an estimator of $\theta$ with possibly nonzero bias $B(\theta)=E_{\theta} \tilde{\theta}-\theta$. Suppose the function $B$ is monotone increasing on $(0, \infty)$. Prove that the mean-squared errors satisfy

  $$E_{\theta}\left(\tilde{\theta}_{n}-\theta\right)^{2} \geqslant E_{\theta}\left(\bar{X}_{n}-\theta\right)^{2} \text { for all } \theta \in \Theta$$

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• Paper 4, Section II, J

  Principles of Statistics

  Part II, 2016

  Consider a decision problem with parameter space $\Theta$. Define the concepts of a Bayes decision rule $\delta_{\pi}$ and of a least favourable prior.

  Suppose $\pi$ is a prior distribution on $\Theta$ such that the Bayes risk of the Bayes rule equals $\sup _{\theta \in \Theta} R\left(\delta_{\pi}, \theta\right)$, where $R(\delta, \theta)$ is the risk function associated to the decision problem. Prove that $\delta_{\pi}$ is least favourable.

  Now consider a random variable $X$ arising from the binomial distribution $\operatorname{Bin}(n, \theta)$, where $\theta \in \Theta=[0,1]$. Construct a least favourable prior for the squared risk $R(\delta, \theta)=E_{\theta}(\delta(X)-\theta)^{2}$. [You may use without proof the fact that the Bayes rule for quadratic risk is given by the posterior mean.]

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• Paper 1, Section II, $\mathbf{2 7 J}$

  Principles of Statistics

  Part II, 2016

  Derive the maximum likelihood estimator $\hat{\theta}_{n}$ based on independent observations $X_{1}, \ldots, X_{n}$ that are identically distributed as $N(\theta, 1)$, where the unknown parameter $\theta$ lies in the parameter space $\Theta=\mathbb{R}$. Find the limiting distribution of $\sqrt{n}\left(\widehat{\theta}_{n}-\theta\right)$ as $n \rightarrow \infty$.

  Now define

  $$\begin{array}{rll} \tilde{\theta}_{n} & =\widehat{\theta}_{n} & \text { whenever }\left|\widehat{\theta}_{n}\right|>n^{-1 / 4}, \\ & =0 & \text { otherwise, } \end{array}$$

  $$\begin{aligned} & =0 \text { otherwise, } \end{aligned}$$

  and find the limiting distribution of $\sqrt{n}\left(\tilde{\theta}_{n}-\theta\right)$ as $n \rightarrow \infty$.

  Calculate

  $$\lim _{n \rightarrow \infty} \sup _{\theta \in \Theta} n E_{\theta}\left(T_{n}-\theta\right)^{2}$$

  for the choices $T_{n}=\widehat{\theta}_{n}$ and $T_{n}=\widetilde{\theta}_{n}$. Based on the above findings, which estimator $T_{n}$ of $\theta$ would you prefer? Explain your answer.

  [Throughout, you may use standard facts of stochastic convergence, such as the central limit theorem, provided they are clearly stated.]

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• Paper 3, Section II, J

  Probability and Measure

  Part II, 2016

  (a) Define the Borel $\sigma$-algebra $\mathcal{B}$ and the Borel functions.

  (b) Give an example with proof of a set in $[0,1]$ which is not Lebesgue measurable.

  (c) The Cantor set $\mathcal{C}$ is given by

  $$\mathcal{C}=\left\{\sum_{k=1}^{\infty} \frac{a_{k}}{3^{k}}:\left(a_{k}\right) \text { is a sequence with } a_{k} \in\{0,2\} \text { for all } k\right\}$$

  (i) Explain why $\mathcal{C}$ is Lebesgue measurable.

  (ii) Compute the Lebesgue measure of $\mathcal{C}$.

  (iii) Is every subset of $\mathcal{C}$ Lebesgue measurable?

  (iv) Let $f:[0,1] \rightarrow \mathcal{C}$ be the function given by

  $$f(x)=\sum_{k=1}^{\infty} \frac{2 a_{k}}{3^{k}} \quad \text { where } \quad a_{k}=\left\lfloor 2^{k} x\right\rfloor-2\left\lfloor 2^{k-1} x\right\rfloor$$

  Explain why $f$ is a Borel function.

  (v) Using the previous parts, prove the existence of a Lebesgue measurable set which is not Borel.

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• Paper 4, Section II, J

  Probability and Measure

  Part II, 2016

  Give the definitions of the convolution $f * g$ and of the Fourier transform $\widehat{f}$ of $f$, and show that $\widehat{f * g}=\widehat{f} \widehat{g}$. State what it means for Fourier inversion to hold for a function $f$.

  State the Plancherel identity and compute the $L^{2}$ norm of the Fourier transform of the function $f(x)=e^{-x} \mathbf{1}_{[0,1]}$.

  Suppose that $\left(f_{n}\right), f$ are functions in $L^{1}$ such that $f_{n} \rightarrow f$ in $L^{1}$ as $n \rightarrow \infty$. Show that $\widehat{f}_{n} \rightarrow \widehat{f}$ uniformly.

  Give the definition of weak convergence, and state and prove the Central Limit Theorem.

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• Paper 2, Section II, J

  Probability and Measure

  Part II, 2016

  (a) State Jensen's inequality. Give the definition of $\|\cdot\|_{L^{p}}$ and the space $L^{p}$ for $1<p<\infty$. If $\|f-g\|_{L^{p}}=0$, is it true that $f=g$ ? Justify your answer. State and prove Hölder's inequality using Jensen's inequality.

  (b) Suppose that $(E, \mathcal{E}, \mu)$ is a finite measure space. Show that if $1<q<p$ and $f \in L^{p}(E)$ then $f \in L^{q}(E)$. Give the definition of $\|\cdot\|_{L^{\infty}}$ and show that $\|f\|_{L^{p}} \rightarrow\|f\|_{L^{\infty}}$ as $p \rightarrow \infty$.

  (c) Suppose that $1<q<p<\infty$. Show that if $f$ belongs to both $L^{p}(\mathbb{R})$ and $L^{q}(\mathbb{R})$, then $f \in L^{r}(\mathbb{R})$ for any $r \in[q, p]$. If $f \in L^{p}(\mathbb{R})$, must we have $f \in L^{q}(\mathbb{R})$ ? Give a proof or a counterexample.

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• Paper 1, Section II, J

  Probability and Measure

  Part II, 2016

  Throughout this question $(E, \mathcal{E}, \mu)$ is a measure space and $\left(f_{n}\right), f$ are measurable functions.

  (a) Give the definitions of pointwise convergence, pointwise a.e. convergence, and convergence in measure.

  (b) If $f_{n} \rightarrow f$ pointwise a.e., does $f_{n} \rightarrow f$ in measure? Give a proof or a counterexample.

  (c) If $f_{n} \rightarrow f$ in measure, does $f_{n} \rightarrow f$ pointwise a.e.? Give a proof or a counterexample.

  (d) Now suppose that $(E, \mathcal{E})=([0,1], \mathcal{B}([0,1]))$ and that $\mu$ is Lebesgue measure on $[0,1]$. Suppose $\left(f_{n}\right)$ is a sequence of Borel measurable functions on $[0,1]$ which converges pointwise a.e. to $f$.

  (i) For each $n, k$ let $E_{n, k}=\bigcup_{m \geqslant n}\left\{x:\left|f_{m}(x)-f(x)\right|>1 / k\right\}$. Show that $\lim _{n \rightarrow \infty} \mu\left(E_{n, k}\right)=0$ for each $k \in \mathbb{N}$.

  (ii) Show that for every $\epsilon>0$ there exists a set $A$ with $\mu(A)<\epsilon$ so that $f_{n} \rightarrow f$ uniformly on $[0,1] \backslash A$.

  (iii) Does (ii) hold with $[0,1]$ replaced by $\mathbb{R}$ ? Give a proof or a counterexample.

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• Paper 3, Section II, I

  Representation Theory

  Part II, 2016

  (a) Let the finite group $G$ act on a finite set $X$ and let $\pi$ be the permutation character. If $G$ is 2 -transitive on $X$, show that $\pi=1_{G}+\chi$, where $\chi$ is an irreducible character of $G$.

  (b) Let $n \geqslant 4$, and let $G$ be the symmetric group $S_{n}$ acting naturally on the set $X=\{1, \ldots, n\}$. For any integer $r \leqslant n / 2$, write $X_{r}$ for the set of all $r$-element subsets of $X$, and let $\pi_{r}$ be the permutation character of the action of $G$ on $X_{r}$. Compute the degree of $\pi_{r}$. If $0 \leqslant \ell \leqslant k \leqslant n / 2$, compute the character inner product $\left\langle\pi_{k}, \pi_{\ell}\right\rangle$.

  Let $m=n / 2$ if $n$ is even, and $m=(n-1) / 2$ if $n$ is odd. Deduce that $S_{n}$ has distinct irreducible characters $\chi^{(n)}=1_{G}, \chi^{(n-1,1)}, \chi^{(n-2,2)}, \ldots, \chi^{(n-m, m)}$ such that for all $r \leqslant m$,

  $$\pi_{r}=\chi^{(n)}+\chi^{(n-1,1)}+\chi^{(n-2,2)}+\cdots+\chi^{(n-r, r)}$$

  (c) Let $\Omega$ be the set of all ordered pairs $(i, j)$ with $i, j \in\{1,2, \ldots, n\}$ and $i \neq j$. Let $S_{n}$ act on $\Omega$ in the obvious way. Write $\pi^{(n-2,1,1)}$ for the permutation character of $S_{n}$ in this action. By considering inner products, or otherwise, prove that

  $$\pi^{(n-2,1,1)}=1+2 \chi^{(n-1,1)}+\chi^{(n-2,2)}+\psi$$

  where $\psi$ is an irreducible character. Calculate the degree of $\psi$, and calculate its value on the elements $\left(\begin{array}{ll}1 & 2\end{array}\right)$ and $\left(\begin{array}{lll}1 & 2 & 3)\end{array}\right)$ of $S_{n}$.

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• Paper 2, Section II, I

  Representation Theory

  Part II, 2016

  Show that the 1-dimensional (complex) characters of a finite group $G$ form a group under pointwise multiplication. Denote this group by $\widehat{G}$. Show that if $g \in G$, the map $\chi \mapsto \chi(g)$ from $\widehat{G}$ to $\mathbb{C}$ is a character of $\widehat{G}$, hence an element of $\widehat{G}$. What is the kernel of the $\operatorname{map} G \rightarrow \widehat{\widehat{G}}$ ?

  Show that if $G$ is abelian the map $G \rightarrow \widehat{\widehat{G}}$ is an isomorphism. Deduce, from the structure theorem for finite abelian groups, that the groups $G$ and $\widehat{G}$ are isomorphic as abstract groups.

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• Paper 4, Section II, I

  Representation Theory

  Part II, 2016

  Let $N$ be a proper normal subgroup of a finite group $G$ and let $U$ be an irreducible complex representation of $G$. Show that either $U$ restricted to $N$ is a sum of copies of a single irreducible representation of $N$, or else $U$ is induced from an irreducible representation of some proper subgroup of $G$.

  Recall that a $p$-group is a group whose order is a power of the prime number $p$. Deduce, by induction on the order of the group, or otherwise, that every irreducible complex representation of a $p$-group is induced from a 1-dimensional representation of some subgroup.

  [You may assume that a non-abelian $p$-group $G$ has an abelian normal subgroup which is not contained in the centre of $G$.]

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• Paper 1, Section II, I

  Representation Theory

  Part II, 2016

  Let $N$ be a normal subgroup of the finite group $G$. Explain how a (complex) representation of $G / N$ gives rise to an associated representation of $G$, and briefly describe which representations of $G$ arise this way.

  Let $G$ be the group of order 54 which is given by

  $$G=\left\langle a, b: a^{9}=b^{6}=1, b^{-1} a b=a^{2}\right\rangle$$

  Find the conjugacy classes of $G$. By observing that $N_{1}=\langle a\rangle$ and $N_{2}=\left\langle a^{3}, b^{2}\right\rangle$ are normal in $G$, or otherwise, construct the character table of $G$.

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• Paper 3, Section II, H

  Riemann Surfaces

  Part II, 2016

  Let $f$ be a non-constant elliptic function with respect to a lattice $\Lambda \subset \mathbb{C}$. Let $P \subset \mathbb{C}$ be a fundamental parallelogram and let the degree of $f$ be $n$. Let $a_{1}, \ldots, a_{n}$ denote the zeros of $f$ in $P$, and let $b_{1}, \ldots, b_{n}$ denote the poles (both with possible repeats). By considering the integral (if required, also slightly perturbing $P$ )

  $$\frac{1}{2 \pi i} \int_{\partial P} z \frac{f^{\prime}(z)}{f(z)} d z$$

  show that

  $$\sum_{j=1}^{n} a_{j}-\sum_{j=1}^{n} b_{j} \in \Lambda$$

  Let $\wp(z)$ denote the Weierstrass $\wp$-function with respect to $\Lambda$. For $v, w \notin \Lambda$ with $\wp(v) \neq \wp(w)$ we set

  $$f(z)=\operatorname{det}\left(\begin{array}{ccc} 1 & 1 & 1 \\ \wp(z) & \wp(v) & \wp(w) \\ \wp^{\prime}(z) & \wp^{\prime}(v) & \wp^{\prime}(w) \end{array}\right)$$

  an elliptic function with periods $\Lambda$. Suppose $z \notin \Lambda, z-v \notin \Lambda$ and $z-w \notin \Lambda$. Prove that $f(z)=0$ if and only if $z+v+w \in \Lambda$. [You may use standard properties of the Weierstrass $\wp$-function provided they are clearly stated.]

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• Paper 2, Section II, H

  Riemann Surfaces

  Part II, 2016

  Suppose that $f: \mathbb{C} / \Lambda_{1} \rightarrow \mathbb{C} / \Lambda_{2}$ is a holomorphic map of complex tori, and let $\pi_{j}$ denote the projection map $\mathbb{C} \rightarrow \mathbb{C} / \Lambda_{j}$ for $j=1,2$. Show that there is a holomorphic map $F: \mathbb{C} \rightarrow \mathbb{C}$ such that $\pi_{2} F=f \pi_{1} .$

  Prove that $F(z)=\lambda z+\mu$ for some $\lambda, \mu \in \mathbb{C}$. Hence deduce that two complex tori $\mathbb{C} / \Lambda_{1}$ and $\mathbb{C} / \Lambda_{2}$ are conformally equivalent if and only if the lattices are related by $\Lambda_{2}=\lambda \Lambda_{1}$ for some $\lambda \in \mathbb{C}^{*}$.

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• Paper 1, Section II, H

  Riemann Surfaces

  Part II, 2016

  (a) Let $f: R \rightarrow S$ be a non-constant holomorphic map between Riemann surfaces. Prove that $f$ takes open sets of $R$ to open sets of $S$.

  (b) Let $U$ be a simply connected domain strictly contained in $\mathbb{C}$. Is there a conformal equivalence between $U$ and $\mathbb{C}$ ? Justify your answer.

  (c) Let $R$ be a compact Riemann surface and $A \subset R$ a discrete subset. Given a non-constant holomorphic function $f: R \backslash A \rightarrow \mathbb{C}$, show that $f(R \backslash A)$ is dense in $\mathbb{C}$.

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• Paper 2, Section I, K

  Statistical Modelling

  Part II, 2016

  Define an exponential dispersion family. Prove that the range of the natural parameter, $\Theta$, is an open interval. Derive the mean and variance as a function of the log normalizing constant.

  [Hint: Use the convexity of $e^{x}$, i.e. $e^{p x+(1-p) y} \leqslant p e^{x}+(1-p) e^{y}$ for all $\left.p \in[0,1] .\right]$

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• Paper 4, Section I, K

  Statistical Modelling

  Part II, 2016

  (a) Let $Y_{i}=x_{i}^{\top} \beta+\varepsilon_{i}$ where $\varepsilon_{i}$ for $i=1, \ldots, n$ are independent and identically distributed. Let $Z_{i}=I\left(Y_{i}<0\right)$ for $i=1, \ldots, n$, and suppose that these variables follow a binary regression model with the complementary log-log link function $g(\mu)=$ $\log (-\log (1-\mu))$. What is the probability density function of $\varepsilon_{1}$ ?

  (b) The Newton-Raphson algorithm can be applied to compute the MLE, $\hat{\beta}$, in certain GLMs. Starting from $\beta^{(0)}=0$, we let $\beta^{(t+1)}$ be the maximizer of the quadratic approximation of the log-likelihood $\ell(\beta ; Y)$ around $\beta^{(t)}$ :

  $$\ell(\beta ; Y) \approx \ell\left(\beta^{(t)} ; Y\right)+\left(\beta-\beta^{(t)}\right)^{\top} D \ell\left(\beta^{(t)} ; Y\right)+\left(\beta-\beta^{(t)}\right)^{\top} D^{2} \ell\left(\beta^{(t)} ; Y\right)\left(\beta-\beta^{(t)}\right),$$

  where $D \ell$ and $D^{2} \ell$ are the gradient and Hessian of the log-likelihood. What is the difference between this algorithm and Iterative Weighted Least Squares? Why might the latter be preferable?

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• Paper 3, Section I, K

  Statistical Modelling

  Part II, 2016

  The $R$ command

  $>\operatorname{boxcox}($ rainfall $\sim$ month+elnino+month:elnino)

  performs a Box-Cox transform of the response at several values of the parameter $\lambda$, and produces the following plot:

  

  We fit two linear models and obtain the Q-Q plots for each fit, which are shown below in no particular order:

  

  Define the variable on the $y$-axis in the output of boxcox, and match each Q-Q plot to one of the models.

  After choosing the model fit.2, the researcher calculates Cook's distance for the $i$ th sample, which has high leverage, and compares it to the upper $0.01$-point of an $F_{p, n-p}$ distribution, because the design matrix is of size $n \times p$. Provide an interpretation of this comparison in terms of confidence sets for $\hat{\beta}$. Is this confidence statement exact?

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• Paper 1, Section I, K

  Statistical Modelling

  Part II, 2016

  The body mass index (BMI) of your closest friend is a good predictor of your own BMI. A scientist applies polynomial regression to understand the relationship between these two variables among 200 students in a sixth form college. The $R$ commands

  $>$ fit. $1<-\operatorname{lm}(B M I \sim$ poly $($ friendBMI , 2, raw=T $))$

  $>$ fit. $2<-\operatorname{lm}(B M I \sim$ poly $($ friendBMI, 3, raw $=\mathrm{T}))$

  fit the models $Y=\beta_{0}+\beta_{1} X+\beta_{2} X^{2}+\varepsilon$ and $Y=\beta_{0}+\beta_{1} X+\beta_{2} X^{2}+\beta_{3} X^{3}+\varepsilon$, respectively, with $\varepsilon \sim N\left(0, \sigma^{2}\right)$ in each case.

  Setting the parameters raw to FALSE:

  $>$ fit. $3<-\operatorname{lm}(B M I \sim$ poly $($ friendBMI , 2, raw=F $)$ )

  $>$ fit. $4<-\operatorname{lm}(\mathrm{BMI} \sim$ poly $($ friendBMI, 3, raw $=\mathrm{F}))$

  fits the models $Y=\beta_{0}+\beta_{1} P_{1}(X)+\beta_{2} P_{2}(X)+\varepsilon$ and $Y=\beta_{0}+\beta_{1} P_{1}(X)+\beta_{2} P_{2}(X)+$ $\beta_{3} P_{3}(X)+\varepsilon$, with $\varepsilon \sim N\left(0, \sigma^{2}\right)$. The function $P_{i}$ is a polynomial of degree $i$. Furthermore, the design matrix output by the function poly with raw=F satisfies:

  $>t($ poly $($ friendBMI, 3, raw $=F)) \% * \%$ poly $(a, 3$, raw $=F)$

  $\begin{array}{rrrr}1 & 2 & 3 \\ 1 & 1.000000 e+00 & 1.288032 \mathrm{e}-16 & 3.187554 \mathrm{e}-17 \\ 2 & 1.288032 \mathrm{e}-16 & 1.000000 \mathrm{e}+00 & -6.201636 \mathrm{e}-17 \\ 3 & 3.187554 \mathrm{e}-17 & -6.201636 \mathrm{e}-17 & 1.000000 \mathrm{e}+00\end{array}$

  How does the variance of $\hat{\beta}$ differ in the models $f i t .2$ and $f i t .4$ ? What about the variance of the fitted values $\hat{Y}=X \hat{\beta}$ ? Finally, consider the output of the commands

  $>\operatorname{anova}$ (fit.1,fit.2)

  anova(fit.3,fit.4)

  Define the test statistic computed by this function and specify its distribution. Which command yields a higher statistic?

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• Paper 4, Section II, K

  Statistical Modelling

  Part II, 2016

  For 31 days after the outbreak of the 2014 Ebola epidemic, the World Health Organization recorded the number of new cases per day in 60 hospitals in West Africa. Researchers are interested in modelling $Y_{i j}$, the number of new Ebola cases in hospital $i$ on day $j \geqslant 2$, as a function of several covariates:

  • lab: a Boolean factor for whether the hospital has laboratory facilities,

  • casesBefore: number of cases at the hospital on the previous day,

  • urban: a Boolean factor indicating an urban area,

  • country: a factor with three categories, Guinea, Liberia, and Sierra Leone,

  • numDoctors: number of doctors at the hospital,

  • tradBurials: a Boolean factor indicating whether traditional burials are common in the region.

  Consider the output of the following $R$ code (with some lines omitted):

  fit. 1 <- glm(newCases lab+casesBefore+urban+country+numDoctors+tradBurials,

  • data=ebola, family=poisson)

  $>$ summary (fit.1)

  Coefficients:

  Estimate Std. Error z value $\operatorname{Pr}(>|z|)$

  $\begin{array}{lllll}\text { labTRUE } & 0.094731 & 0.050322 & 1.882 & 0.0598 \\ \text { (Intercept) } & 0.011298 & 0.049498 & 0.228 & 0.8195\end{array}$

  casesBefore $\quad 0.324744 \quad 0.007752 \quad 41.891<2 \mathrm{e}-16 * * *$

  $\begin{array}{llllll}\text { urbanTRUE } & -0.091554 & 0.088212 & -1.038 & 0.2993\end{array}$

  countryLiberia $\quad 0.088490 \quad 0.034119 \quad 2.594 \quad 0.0095 * *$

  countrySierra Leone $-0.197474 \quad 0.036969-5.3429 .21 \mathrm{e}-08 * * *$

  numDoctors $\quad-0.020819 \quad 0.004658-4.4707 .83 \mathrm{e}-06 * * *$

  tradBurialstrUE $\quad 0.054296 \quad 0.031676 \quad 1.714 \quad 0.0865 .$

  Signif. codes: $0{ }^{\prime} * * *, 0.0011^{\prime} * *, 0.01, *, 0.05, \ldots 0.1,1$

  (a) Would you conclude based on the $z$-tests that an urban setting does not affect the rate of infection?

  (b) Explain how you would predict the total number of new cases that the researchers will record in Sierra Leone on day 32 .

  We fit a new model which includes an interaction term, and compute a test statistic using the code:

  $>$ fit. $2<-$ glm (newCases $\sim$ casesBefore+country+country:casesBefore+numDoctors,

  • data=ebola, family=poisson)

  fit. 2 deviance - fit.1$deviance

  [1] $3.016138$

  (c) What is the distribution of the statistic computed in the last line?

  (d) Under what conditions is the deviance of each model approximately chi-squared?

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• Paper 1, Section II, K

  Statistical Modelling

  Part II, 2016

  (a) Let $Y$ be an $n$-vector of responses from the linear model $Y=X \beta+\varepsilon$, with $\beta \in \mathbb{R}^{p}$. The internally studentized residual is defined by

  $$s_{i}=\frac{Y_{i}-x_{i}^{\top} \hat{\beta}}{\tilde{\sigma} \sqrt{1-p_{i}}},$$

  where $\hat{\beta}$ is the least squares estimate, $p_{i}$ is the leverage of sample $i$, and

  $$\tilde{\sigma}^{2}=\frac{\|Y-X \hat{\beta}\|_{2}^{2}}{(n-p)} .$$

  Prove that the joint distribution of $s=\left(s_{1}, \ldots, s_{n}\right)^{\top}$ is the same in the following two models: (i) $\varepsilon \sim N(0, \sigma I)$, and (ii) $\varepsilon \mid \sigma \sim N(0, \sigma I)$, with $1 / \sigma \sim \chi_{\nu}^{2}$ (in this model, $\varepsilon_{1}, \ldots, \varepsilon_{n}$ are identically $t_{\nu}$-distributed). [Hint: A random vector $Z$ is spherically symmetric if for any orthogonal matrix $H, H Z \stackrel{d}{=} Z$. If $Z$ is spherically symmetric and a.s. nonzero, then $Z /\|Z\|_{2}$ is a uniform point on the sphere; in addition, any orthogonal projection of $Z$ is also spherically symmetric. A standard normal vector is spherically symmetric.]

  (b) A social scientist regresses the income of 120 Cambridge graduates onto 20 answers from a questionnaire given to the participants in their first year. She notices one questionnaire with very unusual answers, which she suspects was due to miscoding. The sample has a leverage of $0.8$. To check whether this sample is an outlier, she computes its externally studentized residual,

  $$t_{i}=\frac{Y_{i}-x_{i}^{\top} \hat{\beta}}{\tilde{\sigma}_{(i)} \sqrt{1-p_{i}}}=4.57,$$

  where $\tilde{\sigma}_{(i)}$ is estimated from a fit of all samples except the one in question, $\left(x_{i}, Y_{i}\right)$. Is this a high leverage point? Can she conclude this sample is an outlier at a significance level of $5 \%$ ?

  (c) After examining the following plot of residuals against the response, the investigator calculates the externally studentized residual of the participant denoted by the black dot, which is $2.33$. Can she conclude this sample is an outlier with a significance level of $5 \%$ ?

  

  Part II, $2016 \quad$ List of Questions

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• Paper 1, Section II, C

  Statistical Physics

  Part II, 2016

  Consider an ideal quantum gas with one-particle states $|i\rangle$ of energy $\epsilon_{i}$. Let $p_{i}^{\left(n_{i}\right)}$ denote the probability that state $|i\rangle$ is occupied by $n_{i}$ particles. Here, $n_{i}$ can take the values 0 or 1 for fermions and any non-negative integer for bosons. The entropy of the gas is given by

  $$S=-k_{B} \sum_{i} \sum_{n_{i}} p_{i}^{\left(n_{i}\right)} \ln p_{i}^{\left(n_{i}\right)}$$

  (a) Write down the constraints that must be satisfied by the probabilities if the average energy $\langle E\rangle$ and average particle number $\langle N\rangle$ are kept at fixed values.

  Show that if $S$ is maximised then

  $$p_{i}^{\left(n_{i}\right)}=\frac{1}{\mathcal{Z}_{i}} e^{-\left(\beta \epsilon_{i}+\gamma\right) n_{i}}$$

  where $\beta$ and $\gamma$ are Lagrange multipliers. What is $\mathcal{Z}_{i}$ ?

  (b) Insert these probabilities $p_{i}^{\left(n_{i}\right)}$ into the expression for $S$, and combine the result with the first law of thermodynamics to find the meaning of $\beta$ and $\gamma$.

  (c) Calculate the average occupation number $\left\langle n_{i}\right\rangle=\sum_{n_{i}} n_{i} p_{i}^{\left(n_{i}\right)}$ for a gas of fermions.

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• Paper 3, Section II, C

  Statistical Physics

  Part II, 2016

  (a) Consider an ideal gas consisting of $N$ identical classical particles of mass $m$ moving freely in a volume $V$ with Hamiltonian $H=|\mathbf{p}|^{2} / 2 m$. Show that the partition function of the gas has the form

  $$Z_{\text {ideal }}=\frac{V^{N}}{\lambda^{3 N} N !}$$

  and find $\lambda$ as a function of the temperature $T$.

  [You may assume that $\int_{-\infty}^{\infty} e^{-a x^{2}} d x=\sqrt{\pi / a}$ for $a>0$.]

  (b) A monatomic gas of interacting particles is a modification of an ideal gas where any pair of particles with separation $r$ interact through a potential energy $U(r)$. The partition function for this gas can be written as

  $$Z=Z_{\text {ideal }}\left[1+\frac{2 \pi N}{V} \int_{0}^{\infty} f(r) r^{2} d r\right]^{N}$$

  where $f(r)=e^{-\beta U(r)}-1, \quad \beta=1 /\left(k_{B} T\right)$. The virial expansion of the equation of state for small densities $N / V$ is

  $$\frac{p}{k_{B} T}=\frac{N}{V}+B_{2}(T) \frac{N^{2}}{V^{2}}+\mathcal{O}\left(\frac{N^{3}}{V^{3}}\right)$$

  Using the free energy, show that

  $$B_{2}(T)=-2 \pi \int_{0}^{\infty} f(r) r^{2} d r$$

  (c) The Lennard-Jones potential is

  $$U(r)=\epsilon\left(\frac{r_{0}^{12}}{r^{12}}-2 \frac{r_{0}^{6}}{r^{6}}\right)$$

  where $\epsilon$ and $r_{0}$ are positive constants. Find the separation $\sigma$ where $U(\sigma)=0$ and the separation $r_{\min }$ where $U(r)$ has its minimum. Sketch the graph of $U(r)$. Calculate $B_{2}(T)$ for this potential using the approximations

  $$f(r)=e^{-\beta U(r)}-1 \simeq \begin{cases}-1 & \text { for } \quad r<\sigma \\ -\beta U(r) & \text { for } r \geqslant \sigma\end{cases}$$

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• Paper 4, Section II, 33C

  Statistical Physics

  Part II, 2016

  (a) State the first law of thermodynamics. Derive the Maxwell relation

  $$\left(\frac{\partial S}{\partial V}\right)_{T}=\left(\frac{\partial p}{\partial T}\right)_{V}$$

  (b) Consider a thermodynamic system whose energy $E$ at constant temperature $T$ is volume independent, i.e.

  $$\left(\frac{\partial E}{\partial V}\right)_{T}=0$$

  Show that this implies that the pressure has the form $p(T, V)=T f(V)$ for some function $f$.

  (c) For a photon gas inside a cavity of volume $V$, the energy $E$ and pressure $p$ are given in terms of the energy density $U$, which depends only on the temperature $T$, by

  $$E(T, V)=U(T) V, \quad p(T, V)=\frac{1}{3} U(T)$$

  Show that this implies $U(T)=\sigma T^{4}$ where $\sigma$ is a constant. Show that the entropy is

  $$S=\frac{4}{3} \sigma T^{3} V$$

  and calculate the energy $E(S, V)$ and free energy $F(T, V)$ in terms of their respective fundamental variables.

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• Paper 2, Section II, C

  Statistical Physics

  Part II, 2016

  (a) What is meant by the canonical ensemble? Consider a system in the canonical ensemble that can be in states $|n\rangle, n=0,1,2, \ldots$ with energies $E_{n}$. Write down the partition function for this system and the probability $p(n)$ that the system is in state $|n\rangle$. Derive an expression for the average energy $\langle E\rangle$ in terms of the partition function.

  (b) Consider an anharmonic oscillator with energy levels

  $$\hbar \omega\left[\left(n+\frac{1}{2}\right)+\delta\left(n+\frac{1}{2}\right)^{2}\right], \quad n=0,1,2, \ldots$$

  where $\omega$ is a positive constant and $0<\delta \ll 1$ is a small constant. Let the oscillator be in contact with a reservoir at temperature $T$. Show that, to linear order in $\delta$, the partition function $Z_{1}$ for the oscillator is given by

  $$Z_{1}=\frac{c_{1}}{\sinh \frac{x}{2}}\left[1+\delta c_{2} x\left(1+\frac{2}{\sinh ^{2} \frac{x}{2}}\right)\right], \quad x=\frac{\hbar \omega}{k_{B} T}$$

  where $c_{1}$ and $c_{2}$ are constants to be determined. Also show that, to linear order in $\delta$, the average energy of a system of $N$ uncoupled oscillators of this type is given by

  $$\langle E\rangle=\frac{N \hbar \omega}{2}\left\{c_{3} \operatorname{coth} \frac{x}{2}+\delta\left[c_{4}+\frac{c_{5}}{\sinh ^{2} \frac{x}{2}}\left(1-x \operatorname{coth} \frac{x}{2}\right)\right]\right\}$$

  where $c_{3}, c_{4}, c_{5}$ are constants to be determined.

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• Paper 3, Section II, K

  Stochastic Financial Models

  Part II, 2016

  Consider the following two-period market model. There is a risk-free asset which pays interest at rate $r=1 / 4$. There is also a risky stock with prices $\left(S_{t}\right)_{t \in\{0,1,2\}}$ given by

  

  The above diagram should be read as

  $$\mathbb{P}\left(S_{1}=10 \mid S_{0}=7\right)=2 / 3, \quad \mathbb{P}\left(S_{2}=14 \mid S_{1}=10\right)=1 / 2$$

  and so forth.

  (a) Find the risk-neutral probabilities.

  (b) Consider a European put option with strike $K=10$ expiring at time $T=2$. What is the initial no-arbitrage price of the option? How many shares should be held in each period to replicate the payout?

  (c) Now consider an American put option with the same strike and expiration date. Find the optimal exercise policy, assuming immediate exercise is not allowed. Would your answer change if you were allowed to exercise the option at time 0 ?

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• Paper 4, Section II, K

  Stochastic Financial Models

  Part II, 2016

  Let $U$ be concave and strictly increasing, and let $\mathcal{A}$ be a vector space of random variables. For every random variable $Z$ let

  $$F(Z)=\sup _{X \in \mathcal{A}} \mathbb{E}[U(X+Z)]$$

  and suppose there exists a random variable $X_{Z} \in \mathcal{A}$ such that

  $$F(Z)=\mathbb{E}\left[U\left(X_{Z}+Z\right)\right]$$

  For a random variable $Y$, let $\pi(Y)$ be such that $F(Y-\pi(Y))=F(0)$.

  (a) Show that for every constant $a$ we have $\pi(Y+a)=\pi(Y)+a$, and that if $\mathbb{P}\left(Y_{1} \leqslant Y_{2}\right)=1$, then $\pi\left(Y_{1}\right) \leqslant \pi\left(Y_{2}\right)$. Hence show that if $\mathbb{P}(a \leqslant Y \leqslant b)=1$ for constants $a \leqslant b$, then $a \leqslant \pi(Y) \leqslant b .$

  (b) Show that $Y \mapsto \pi(Y)$ is concave, and hence show $t \mapsto \pi(t Y) / t$ is decreasing for $t>0$.

  (c) Assuming $U$ is continuously differentiable, show that $\pi(t Y) / t$ converges as $t \rightarrow 0$, and that there exists a random variable $X_{0}$ such that

  $$\lim _{t \rightarrow 0} \frac{\pi(t Y)}{t}=\frac{\mathbb{E}\left[U^{\prime}\left(X_{0}\right) Y\right]}{\mathbb{E}\left[U^{\prime}\left(X_{0}\right)\right]}$$

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• Paper 2, Section II, K

  Stochastic Financial Models

  Part II, 2016

  In the context of the Black-Scholes model, let $S_{0}$ be the initial price of the stock, and let $\sigma$ be its volatility. Assume that the risk-free interest rate is zero and the stock pays no dividends. Let $\operatorname{EC}\left(S_{0}, K, \sigma, T\right)$ denote the initial price of a European call option with strike $K$ and maturity date $T$.

  (a) Show that the Black-Scholes formula can be written in the form

  $$\mathrm{EC}\left(S_{0}, K, \sigma, T\right)=S_{0} \Phi\left(d_{1}\right)-K \Phi\left(d_{2}\right)$$

  where $d_{1}$ and $d_{2}$ depend on $S_{0}, K, \sigma$ and $T$, and $\Phi$ is the standard normal distribution function.

  (b) Let $\operatorname{EP}\left(S_{0}, K, \sigma, T\right)$ be the initial price of a put option with strike $K$ and maturity $T$. Show that

  $$\operatorname{EP}\left(S_{0}, K, \sigma, T\right)=\operatorname{EC}\left(S_{0}, K, \sigma, T\right)+K-S_{0}$$

  (c) Show that

  $$\operatorname{EP}\left(S_{0}, K, \sigma, T\right)=\operatorname{EC}\left(K, S_{0}, \sigma, T\right)$$

  (d) Consider a European contingent claim with maturity $T$ and payout

  $$S_{T} I_{\left\{S_{T} \leqslant K\right\}}-K I_{\left\{S_{T}>K\right\}}$$

  Assuming $K>S_{0}$, show that its initial price can be written as $\mathrm{EC}\left(S_{0}, K, \hat{\sigma}, T\right)$ for a volatility parameter $\hat{\sigma}$ which you should express in terms of $S_{0}, K, \sigma$ and $T$.

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• Paper 1, Section II, 28K

  Stochastic Financial Models

  Part II, 2016

  (a) What is a Brownian motion?

  (b) State the Brownian reflection principle. State the Cameron-Martin theorem for Brownian motion with constant drift.

  (c) Let $\left(W_{t}\right)_{t \geqslant 0}$ be a Brownian motion. Show that

  $$\mathbb{P}\left(\max _{0 \leqslant s \leqslant t}\left(W_{s}+a s\right) \leqslant b\right)=\Phi\left(\frac{b-a t}{\sqrt{t}}\right)-e^{2 a b} \Phi\left(\frac{-b-a t}{\sqrt{t}}\right)$$

  where $\Phi$ is the standard normal distribution function.

  (d) Find

  $$\mathbb{P}\left(\max _{u \geqslant t}\left(W_{u}+a u\right) \leqslant b\right)$$

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• Paper 1, Section I, H

  Topics in Analysis

  Part II, 2016

  By considering the function $\mathbb{R}^{n+1} \rightarrow \mathbb{R}$ defined by

  $$R\left(a_{0}, \ldots, a_{n}\right)=\sup _{t \in[-1,1]}\left|\sum_{j=0}^{n} a_{j} t^{j}\right|$$

  or otherwise, show that there exist $K_{n}>0$ and $\delta_{n}>0$ such that

  $$K_{n} \sum_{j=0}^{n}\left|a_{j}\right| \geqslant \sup _{t \in[-1,1]}\left|\sum_{j=0}^{n} a_{j} t^{j}\right| \geqslant \delta_{n} \sum_{j=0}^{n}\left|a_{j}\right|$$

  for all $a_{j} \in \mathbb{R}, 0 \leqslant j \leqslant n$.

  Show, quoting carefully any theorems you use, that we must have $\delta_{n} \rightarrow 0$ as $n \rightarrow \infty$.

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• Paper 2, Section I, H

  Topics in Analysis

  Part II, 2016

  Define what it means for a subset $E$ of $\mathbb{R}^{n}$ to be convex. Which of the following statements about a convex set $E$ in $\mathbb{R}^{n}$ (with the usual norm) are always true, and which are sometimes false? Give proofs or counterexamples as appropriate.

  (i) The closure of $E$ is convex.

  (ii) The interior of $E$ is convex.

  (iii) If $\alpha: \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ is linear, then $\alpha(E)$ is convex.

  (iv) If $f: \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ is continuous, then $f(E)$ is convex.

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• Paper 3, Section I, $2 \mathrm{H}$

  Topics in Analysis

  Part II, 2016

  In the game of 'Chicken', $A$ and $B$ drive fast cars directly at each other. If they both swerve, they both lose 10 status points; if neither swerves, they both lose 100 status points. If one swerves and the other does not, the swerver loses 20 status points and the non-swerver gains 40 status points. Find all the pairs of probabilistic strategies such that, if one driver maintains their strategy, it is not in the interest of the other to change theirs.

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• Paper 4, Section I, H

  Topics in Analysis

  Part II, 2016

  Let $a_{0}, a_{1}, a_{2}, \ldots$ be integers such that there exists an $M$ with $M \geqslant\left|a_{n}\right|$ for all $n$. Show that, if infinitely many of the $a_{n}$ are non-zero, then $\sum_{n=0}^{\infty} \frac{a_{n}}{n !}$ is an irrational number.

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• Paper 2, Section II, H

  Topics in Analysis

  Part II, 2016

  Prove Bernstein's theorem, which states that if $f:[0,1] \rightarrow \mathbb{R}$ is continuous and

  $$f_{m}(t)=\sum_{r=0}^{m}\left(\begin{array}{c} m \\ r \end{array}\right) f(r / m) t^{r}(1-t)^{m-r}$$

  then $f_{m}(t) \rightarrow f(t)$ uniformly on $[0,1]$. [Theorems from probability theory may be used without proof provided they are clearly stated.]

  Deduce Weierstrass's theorem on polynomial approximation for any closed interval.

  Proving any results on Chebyshev polynomials that you need, show that, if $g:[0, \pi] \rightarrow \mathbb{R}$ is continuous and $\epsilon>0$, then we can find an $N \geqslant 0$ and $a_{j} \in \mathbb{R}$, for $0 \leqslant j \leqslant N$, such that

  $$\left|g(t)-\sum_{j=0}^{N} a_{j} \cos j t\right| \leqslant \epsilon$$

  for all $t \in[0, \pi]$. Deduce that $\int_{0}^{\pi} g(t) \cos n t d t \rightarrow 0$ as $n \rightarrow \infty$.

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• Paper 4, Section II, H

  Topics in Analysis

  Part II, 2016

  Explain briefly how a positive irrational number $x$ gives rise to a continued fraction

  $$a_{0}+\frac{1}{a_{1}+\frac{1}{a_{2}+\frac{1}{a_{3}+\ldots}}}$$

  with the $a_{j}$ non-negative integers and $a_{j} \geqslant 1$ for $j \geqslant 1$.

  Show that, if we write

  $$\left(\begin{array}{ll} p_{n} & p_{n-1} \\ q_{n} & q_{n-1} \end{array}\right)=\left(\begin{array}{cc} a_{0} & 1 \\ 1 & 0 \end{array}\right)\left(\begin{array}{cc} a_{1} & 1 \\ 1 & 0 \end{array}\right) \cdots\left(\begin{array}{cc} a_{n-1} & 1 \\ 1 & 0 \end{array}\right)\left(\begin{array}{cc} a_{n} & 1 \\ 1 & 0 \end{array}\right)$$

  then

  $$\frac{p_{n}}{q_{n}}=a_{0}+\frac{1}{a_{1}+\frac{1}{a_{2}+\frac{1}{a_{n-1}+\frac{1}{a_{n}}}}}$$

  for $n \geqslant 0$.

  Use the observation [which need not be proved] that $x$ lies between $p_{n} / q_{n}$ and $p_{n+1} / q_{n+1}$ to show that

  $$\left|p_{n} / q_{n}-x\right| \leqslant 1 / q_{n} q_{n+1}$$

  Show that $q_{n} \geqslant F_{n}$ where $F_{n}$ is the $n$th Fibonacci number (thus $F_{0}=F_{1}=1$, $\left.F_{n+2}=F_{n+1}+F_{n}\right)$, and conclude that

  $$\left|\frac{p_{n}}{q_{n}}-x\right| \leqslant \frac{1}{F_{n} F_{n+1}}$$

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• Paper 4, Section II, D

  Waves

  Part II, 2016

  A duck swims at a constant velocity $(-V, 0)$, where $V>0$, on the surface of infinitely deep water. Surface tension can be neglected, and the dispersion relation for the linear surface water waves (relative to fluid at rest) is $\omega^{2}=g|\mathbf{k}|$. Show that the wavevector $\mathbf{k}$ of a plane harmonic wave that is steady in the duck's frame, i.e. of the form

  $$\operatorname{Re}\left[A e^{i\left(k_{1} x^{\prime}+k_{2} y\right)}\right]$$

  where $x^{\prime}=x+V t$ and $y$ are horizontal coordinates relative to the duck, satisfies

  $$\left(k_{1}, k_{2}\right)=\frac{g}{V^{2}} \sqrt{p^{2}+1}(1, p)$$

  where $\hat{\mathbf{k}}=(\cos \phi, \sin \phi)$ and $p=\tan \phi$. [You may assume that $|\phi|<\pi / 2 .$ ]

  Assume that the wave pattern behind the duck can be regarded as a Fourier superposition of such steady waves, i.e., the surface elevation $\eta$ at $\left(x^{\prime}, y\right)=R(\cos \theta, \sin \theta)$ has the form

  $$\eta=\operatorname{Re} \int_{-\infty}^{\infty} A(p) e^{i \lambda h(p ; \theta)} \mathrm{d} p \quad \text { for }|\theta|<\frac{1}{2} \pi$$

  where

  $$\lambda=\frac{g R}{V^{2}}, \quad h(p ; \theta)=\sqrt{p^{2}+1}(\cos \theta+p \sin \theta)$$

  Show that, in the limit $\lambda \rightarrow \infty$ at fixed $\theta$ with $0<\theta<\cot ^{-1}(2 \sqrt{2})$,

  $$\eta \sim \sqrt{\frac{2 \pi}{\lambda}} \operatorname{Re}\left\{\frac{A\left(p_{+}\right)}{\sqrt{h_{p p}\left(p_{+} ; \theta\right)}} e^{i\left(\lambda h\left(p_{+} ; \theta\right)+\frac{1}{4} \pi\right)}+\frac{A\left(p_{-}\right)}{\sqrt{-h_{p p}\left(p_{-} ; \theta\right)}} e^{i\left(\lambda h\left(p_{-} ; \theta\right)-\frac{1}{4} \pi\right)}\right\},$$

  where

  $$p_{\pm}=-\frac{1}{4} \cot \theta \pm \frac{1}{4} \sqrt{\cot ^{2} \theta-8}$$

  and $h_{p p}$ denotes $\partial^{2} h / \partial p^{2}$. Briefly interpret this result in terms of what is seen.

  Without doing detailed calculations, briefly explain what is seen as $\lambda \rightarrow \infty$ at fixed $\theta$ with $\cot ^{-1}(2 \sqrt{2})<\theta<\pi / 2$. Very briefly comment on the case $\theta=\cot ^{-1}(2 \sqrt{2})$.

  [Hint: You may find the following results useful.

  $$\begin{gathered} h_{p}=\left\{p \cos \theta+\left(2 p^{2}+1\right) \sin \theta\right\}\left(p^{2}+1\right)^{-1 / 2} \\ \left.h_{p p}=(\cos \theta+4 p \sin \theta)\left(p^{2}+1\right)^{-1 / 2}-\left\{p \cos \theta+\left(2 p^{2}+1\right) \sin \theta\right\} p\left(p^{2}+1\right)^{-3 / 2} \cdot\right] \end{gathered}$$

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• Paper 2, Section II, 37D

  Waves

  Part II, 2016

  Starting from the equations for one-dimensional unsteady flow of a perfect gas at constant entropy, show that the Riemann invariants

  $$R_{\pm}=u \pm \frac{2\left(c-c_{0}\right)}{\gamma-1}$$

  are constant on characteristics $C_{\pm}$given by $d x / d t=u \pm c$, where $u(x, t)$ is the speed of the gas, $c(x, t)$ is the local speed of sound, $c_{0}$ is a constant and $\gamma>1$ is the exponent in the adiabatic equation of state for $p(\rho)$.

  At time $t=0$ the gas occupies $x>0$ and is at rest at uniform density $\rho_{0}$, pressure $p_{0}$ and sound speed $c_{0}$. For $t>0$, a piston initially at $x=0$ has position $x=X(t)$, where

  $$X(t)=-U_{0} t\left(1-\frac{t}{2 t_{0}}\right)$$

  and $U_{0}$ and $t_{0}$ are positive constants. For the case $0<U_{0}<2 c_{0} /(\gamma-1)$, sketch the piston path $x=X(t)$ and the $C_{+}$characteristics in $x \geqslant X(t)$ in the $(x, t)$-plane, and find the time and place at which a shock first forms in the gas.

  Do likewise for the case $U_{0}>2 c_{0} /(\gamma-1)$.

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• Paper 1, Section II, D

  Waves

  Part II, 2016

  Write down the linearised equations governing motion of an inviscid compressible fluid at uniform entropy. Assuming that the velocity is irrotational, show that it may be derived from a velocity potential $\phi(\mathbf{x}, t)$ satisfying the wave equation

  $$\frac{\partial^{2} \phi}{\partial t^{2}}=c_{0}^{2} \nabla^{2} \phi$$

  and identify the wave speed $c_{0}$. Obtain from these linearised equations the energyconservation equation

  $$\frac{\partial E}{\partial t}+\nabla \cdot \mathbf{I}=0$$

  and give expressions for the acoustic-energy density $E$ and the acoustic-energy flux $\mathbf{I}$ in terms of $\phi$.

  Such a fluid occupies a semi-infinite waveguide $x>0$ of square cross-section $0<y<a$, $0<z<a$ bounded by rigid walls. An impenetrable membrane closing the end $x=0$ makes prescribed small displacements to

  $$x=X(y, z, t) \equiv \operatorname{Re}\left[e^{-i \omega t} A(y, z)\right]$$

  where $\omega>0$ and $|A| \ll a, c_{0} / \omega$. Show that the velocity potential is given by

  $$\phi=\operatorname{Re}\left[e^{-i \omega t} \sum_{m=0}^{\infty} \sum_{n=0}^{\infty} \cos \left(\frac{m \pi y}{a}\right) \cos \left(\frac{n \pi z}{a}\right) f_{m n}(x)\right]$$

  where the functions $f_{m n}(x)$, including their amplitudes, are to be determined, with the sign of any square roots specified clearly.

  If $0<\omega<\pi c_{0} / a$, what is the asymptotic behaviour of $\phi$ as $x \rightarrow+\infty$ ? Using this behaviour and the energy-conservation equation averaged over both time and the crosssection, or otherwise, determine the double-averaged energy flux along the waveguide,

  $$\left\langle\overline{I_{x}}\right\rangle(x) \equiv \frac{\omega}{2 \pi a^{2}} \int_{0}^{2 \pi / \omega} \int_{0}^{a} \int_{0}^{a} I_{x}(x, y, z, t) \mathrm{d} y \mathrm{~d} z \mathrm{~d} t$$

  explaining why this is independent of $x$.

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• Paper 3, Section II, $37 \mathrm{D}$

  Waves

  Part II, 2016

  Small disturbances in a homogeneous elastic solid with density $\rho$ and Lamé moduli $\lambda$ and $\mu$ are governed by the equation

  $$\rho \frac{\partial^{2} \mathbf{u}}{\partial t^{2}}=(\lambda+2 \mu) \boldsymbol{\nabla}(\boldsymbol{\nabla} \cdot \mathbf{u})-\mu \boldsymbol{\nabla} \times(\boldsymbol{\nabla} \times \mathbf{u})$$

  where $\mathbf{u}(\mathbf{x}, t)$ is the displacement. Show that a harmonic plane-wave solution

  $$\mathbf{u}=\operatorname{Re}\left[\mathbf{A} e^{i(\mathbf{k} \cdot \mathbf{x}-\omega t)}\right]$$

  must satisfy

  $$\omega^{2} \mathbf{A}=c_{P}^{2} \mathbf{k}(\mathbf{k} \cdot \mathbf{A})-c_{S}^{2} \mathbf{k} \times(\mathbf{k} \times \mathbf{A}),$$

  where the wavespeeds $c_{P}$ and $c_{S}$ are to be identified. Describe mathematically how such plane-wave solutions can be classified into longitudinal $P$-waves and transverse $S V$ - and $S H$-waves (taking the $y$-direction as the vertical direction).

  The half-space $y<0$ is filled with the elastic solid described above, while the slab $0<y<h$ is filled with a homogeneous elastic solid with Lamé moduli $\bar{\lambda}$ and $\bar{\mu}$, and wavespeeds $\bar{c}_{P}$ and $\bar{c}_{S}$. There is a rigid boundary at $y=h$. A harmonic plane $S H$-wave propagates from $y<0$ towards the interface $y=0$, with displacement

  $$\operatorname{Re}\left[A e^{i(\ell x+m y-\omega t)}\right](0,0,1)$$

  How are $\ell, m$ and $\omega$ related? The total displacement in $y<0$ is the sum of $(*)$ and that of the reflected $S H$-wave,

  $$\operatorname{Re}\left[R A e^{i(\ell x-m y-\omega t)}\right](0,0,1)$$

  Write down the form of the displacement in $0<y<h$, and determine the (complex) reflection coefficient $R$. Verify that $|R|=1$ regardless of the parameter values, and explain this physically.

  comment

• Paper 3, Section II, $37 \mathrm{D}$

  Waves

  Part II, 2016

  Small disturbances in a homogeneous elastic solid with density $\rho$ and Lamé moduli $\lambda$ and $\mu$ are governed by the equation

  $$\rho \frac{\partial^{2} \mathbf{u}}{\partial t^{2}}=(\lambda+2 \mu) \boldsymbol{\nabla}(\boldsymbol{\nabla} \cdot \mathbf{u})-\mu \boldsymbol{\nabla} \times(\boldsymbol{\nabla} \times \mathbf{u})$$

  where $\mathbf{u}(\mathbf{x}, t)$ is the displacement. Show that a harmonic plane-wave solution

  $$\mathbf{u}=\operatorname{Re}\left[\mathbf{A} e^{i(\mathbf{k} \cdot \mathbf{x}-\omega t)}\right]$$

  must satisfy

  $$\omega^{2} \mathbf{A}=c_{P}^{2} \mathbf{k}(\mathbf{k} \cdot \mathbf{A})-c_{S}^{2} \mathbf{k} \times(\mathbf{k} \times \mathbf{A}),$$

  where the wavespeeds $c_{P}$ and $c_{S}$ are to be identified. Describe mathematically how such plane-wave solutions can be classified into longitudinal $P$-waves and transverse $S V$ - and $S H$-waves (taking the $y$-direction as the vertical direction).

  The half-space $y<0$ is filled with the elastic solid described above, while the slab $0<y<h$ is filled with a homogeneous elastic solid with Lamé moduli $\bar{\lambda}$ and $\bar{\mu}$, and wavespeeds $\bar{c}_{P}$ and $\bar{c}_{S}$. There is a rigid boundary at $y=h$. A harmonic plane $S H$-wave propagates from $y<0$ towards the interface $y=0$, with displacement

  $$\operatorname{Re}\left[A e^{i(\ell x+m y-\omega t)}\right](0,0,1)$$

  How are $\ell, m$ and $\omega$ related? The total displacement in $y<0$ is the sum of $(*)$ and that of the reflected $S H$-wave,

  $$\operatorname{Re}\left[R A e^{i(\ell x-m y-\omega t)}\right](0,0,1)$$

  Write down the form of the displacement in $0<y<h$, and determine the (complex) reflection coefficient $R$. Verify that $|R|=1$ regardless of the parameter values, and explain this physically.

  comment